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Back button clicked with empty stack but still shown in Recent apps

I am trying to figure out how recent apps interacts with android back button event.
When my app is not opened in the background, if I launched an deeplink and it goes to my simply destination activity. When I clicked on back button, I can see that the activity popped from stack and my app closed due to the activity was the only one in task stack.
my questions,
Even with back button clicked, and the app closed. I still see my app in "recent apps", my impression is that such back button should killed my app since no activity in stack at all. Why still in recent apps?
If I click on the app in recent apps, I observed the same intent was triggered ago (deeplink intent), and why is that? (android's mechanism to cache last intent maybe?)

Back button from my app goes to home screen rather than previous app

I'm trying to debug an issue and I'm not sure what is going on.
My application is launched from application A (this is not mine, someone else's, so I don't have any source code)
When I click the back button on my application, I land on the Android home screen rather than going to the application that launch me. I checked all the flags in the intent that I receive, and everything is turned off.
Now if I launch my application from any other application, then pressing the back button goes back to the application that launch me.
I'm not sure how to debug this issue. Any ideas?

Application A might call finish() once the Intent is sent to start your Activity. Then the Activity in application A that started the Intent to launch your Activity will go through onDestroy() and will be removed from the backstack, so when you press the back key you will go to the home screen if the Activity in Application A was the first Activity launched by Application A

What is the difference between launch app from "recent apps" and tapping app icon

I'm working on large project, so there is some logic for saving application state, and then opening correct activity(fragment) when it comes from background.
But anyway, I've found that if user navigates through my app and then minimize it - android opens it from background in different ways in following cases:
User taps on app icon (behavior: user see home activity, doesn't matter where he was, when application was minimized)
User select app from android "recent apps" (behavior: user see exactly what he saw, when the application was minimized)
So, what is the difference between launching application from background by this two ways? I always thought, that it is the same mechanism, but, obviously, I was wrong.
Thanks for any answers

You should pay atention on the folowing docs Activity and Tasks. In short words: if user start app from recents you will receive onRestart before onStart (without onCreate it means that your app was just "suspended"). You able to save screen state using onSaveInstanceState(). But in general starting from icon and from recents - different application behaviors and you should provide proper code for this ways.
UPD
As described below root cause of unexpected behaviour was additional lunchmode attribute.

From what I experience as an Android user, both are same.
The difference we usually see is how we close the app
Press back button until app close / finish()
On this state no matter how we open the apps it will go to the main screen
Press Home button
On this state depend on the app. If the app does not handle any Activity the app will same with the first state. But if the app handle something like when onPause() the Activity then finish() the apps, then whatever you open with app icon or recent apps will provide the same result.
Correct me if I am wrong

Need clear understanding of Home vs Back events

I would like to know the "idea / use" of the home versus the back button. I mean obviously the home button takes you to the home screen and the back button takes you to the previous screen. What I am trying to understand is what the users / development community expect.
In other words when a user hits home in my application should I handle that event and terminate the application? Is that what the user has grown to expect or just the opposite?
Same for the back button. Do users expect the screen being departed from to be lost much like a web page would be?
What I am getting at is trying to make sure my app behaves consistent with what the user community has grown to expect.
TIA
JB

Home Button will put application in onPause() -> onStop() , again when you relaunch application, the activity will execute the method: onRestart() -> onStart() -> onResume() -> i.e. activity life cycle
and
Back button finishes or kills the current Activity on click of back button and jump back to the previous Activity which is in stack.

Generally I'd say that pressing home is like minimising on a desktop. So I would expect the app to return to where it was.
If it acts like this then I can easily switch between apps if I need to get information from them.
Pressing back should take me backwards through the things I was doing in the app and when I reach the end of the stack it should close.

home button pressing causes an intent that is used by the the launcher app to show itself .
back button is a real event that can be caught within the current app .
in general , home button should hide all apps and go to the launcher , while back button should go back to the previous screen the user saw (finish the activity or dismiss the dialog) , and go to the previous app (task ) in case the current one doesn't have any more screens to show .

The community expects that if you press the home button, your app will go to the background and be resumeable from there. Multitasking is in the core of the Android OS. To finish your activity or app when home button is pressed is unusual to android users.
The back button is when you want to go one step back, like in the browser as you said. If you are know to IOS it will act like when you press the back button there in the top left corner.
Hope this helps

I would like to know the "idea / use" of the home versus the back
button
Let me Discuss for Back Button First.
The back button behavior is inconsistent.
for Users, it is very easy to use.
for Developer, to understand what actually pressing the back button does it isn't so easy.
The back button can perform any of the following actions as Officially said by Android site :
Go back to previous screen (activity)
Dismiss a popup
Terminate a long running process
Close on-screen keyboard
Go to previous page on browser
Exit the running app when on the last activity
Return to previous app when on the last activity and the app was launched through intent from another app
When Problem Occur with BACK button?
An added problem comes when trying to understand when the back button
cancels a running operation and when not. For example, when installing
apps from Android market tapping back takes users to previous screen
and leaves the installation process running in background. I know
there's a rule that back cancels operations that are presented users
as popup progress indicators and any other cases it is used to
navigate. But is that what users expect? Do users have to think before
pressing back to understand what is going to happen?
Home Button
Home button will take users to home screen and swiping in the
multitasking menu will kill the apps if needed. As on ICS on new
phones all the buttons are on-screen the back button could simply get
disabled when user reaches the last activity if the app was launched
from launcher icon. This would make some of the confusion to make away
but still wouldn't solve all the problems.
Finally , Don't Make Users Think Navigation is some what more important. Users should always know where they will endup without thinking.
I don't think there are easy answers for this problem but I think it may help you to find workaround for your problem according to your app.

what exactly android's home button doing?

What I do know is that the current activity get into the onPause() mode, and the home screen activity brought to front.
My confusion starts with situation you can re-open the application from the recent tasks menu.
so what exactly happening when I'm opening the application from the recent tasks manager?
Is the activity that was foreground when the home button pressed is still somewhere in the stack?
Is there more then one activity stack on the same time?

As far as I understand it, there is no real stack (of applications). Is just that your activity has states, so whenever you are pressing the HOME button your activity in your current application just "pauses" like if it was a stand-by state so multiple applications can be in this state as long as the system doesnt require memory and kill the tasks.
So whenever you open the activity from the recent tasks its just telling the application to wake up.

First of all, here is everything you need to know about the concept of the "Up Button":
Navigation with Back and Up and some of the implementation details: Providing Ancestral and Temporal Navigation.
Generally speaking, the Up button lets you navigate up in the application hierarchy, instead of just navigating back in the application(s) back-stack.
For example, if you work with some kind of app and you get the email notification, you can open the mail client by pressing the notification. Then you can go back to you application you were working with by pressing the Back button ( back-stack ) or you can press the Up button in order to go to the mail client's 'parent' activity ( for example from some EmailMessageActivity to EmailHomeActivity ) to work with the mail client application instead of the initial application ( the back-stack usually is cleared then, so you can only go back with the Back button as far as the the Android Home screen ).
The "Recent Tasks" factor is irrelevant and misleading, it's just another way of starting a new Activity.