I was told that the main thing that makes Scrum Scrum is that the team chooses the sprint goal, from amongst the tickets at the top of the backlog, and the team commits to hitting that goal.

In my experience more typical is that the PM chooses a set of tickets and asks the team whether they can commit to them.

According to the constraint above, neither that, nor the “pick from the backlog” process you’re describing would qualify as “Scrum”.