E2E testing a bit confused on what it is suppost to test exactly

So as title say, I'm a bit confused about the E2E concept. When you search for E2E frameworks you get things like Selenium which can automate your browser. Now what I don't quite get is Some definitions of E2E seems to go way beyond simply using Selenium. So for example consider this test: you can add a user to a group on my website. The test might looks like this:

Start chrome  
Go to mypage.com  
Select the group "abc"  
Click add user  
Enter the users name and click "add"  
Verify that the user was added

I've seen some places that say that this is an end to end test. But I've also read that it should not end there. For example in this case, let's say the website uses an API to call the backend and the backend adds the user to a database.

So in this case then it's not enough to simply verify that the user experience was correct but we must also go into the database and check that the user was actually added to the correct group. Should this be covered by the same test? Or should there be separate "database tests" for that kind of verification?