I think the difference is too small to matter enough to deserve further attention or added rules bloat.
Like, people complain about rules bloat all the time. That is exactly what this is. More rules for no reason other than because the existing rule has a disliked name.
Changing the name has really nothing to do with this point. Changing the section title in the book to "Routine and Secret Checks" takes care of the mistaken "passive" issue IMO.
This is about the difference between when someone is trying to notice something and when they simply happen to notice it. When you aren't trying to do something, the effort typically results in less "success".
Truly "passive" (always on in the background) perception should have a lower base value than the 10 given to Routine and Secret Checks score.
In summary:
Routine and Secret Checks (RAW "Passive Checks") default 10 is fine if you want to use it.
(New) Passive Checks default 5 makes a lot more sense as a background use of a skill, something that is innately always on but you aren't
trying to use it necessarily. When you try to use it, you are very likely to do better.
Anyway, you feel active perception versus passive should be close enough to the same to not bother, so I won't bother further.
Frankly, I don't know of anyone who complains of "rules bloat" in 5E!!! Quite the opposite IME. Most people want clearer, more defined rules to avoid confusion. Now, "hit point bloat" and "feature bloat" I have heard about. Perhaps the rules bloat I've missed because I never paid attention to those threads (entirely possible...).