Here's my favorite bands growing up in chronological order: (still love them all)
The Police (1978-84)
The Cult (1984-89)
Red Hot Chili Peppers (1989-96)
Honorable mentions also in chronological order:
Tie: MUSE and TOOL (very much in love with Muse right now)
I haven't seen him live yet, but will the next time he's around!
We have very similar taste Flor.. Loved and grew up with every band you named up to and including System.. Also had a roller hockey team for many, many years named, "South of Heaven".. lol
Genesis (with Gabriel) Gentle Giant, Greenslade, Deep Purple, Mott The Hoople, Wishbone Ash, Yes, Talking Heads, REM, Eagles, Beach Boys, Faust, Kraftwerk, and pretty much everything else!