I realized that I never answered the question you posed! Let's see...well, I feel like insert songs are not really in the spirit of the question (God knows/Lost my music, Fuwa Fuwa Time, Ginga Ensen '05 (okay I know I said this isn't about insert songs but School Rumble has too many banger insert songs to count, just throwing that out there))

A couple from the original Pokémon anime that come to mind:

• 1997~1998-M06 (from when Ash is protecting Pikachu from the flock of Spearow in the first episode AND from the tearful goodbye in "Bye-Bye Butterfree")
• 1997~1998-M02 (not evocative of a single moment I guess, but this is used a ton when our heroes are walking off into the sunset at the end of an episode while the narrator takes us home...instant good vibes)

Shiawase no Sakura-namiki from Cardcaptor Sakura (Sakura dreamily rollerblading under the falling sakura petals)