I'd say I started loving XIV's music in Heavensward. I love pretty much every song in that expansion. I really loved entering Ishgard for the first time and seeing the Heavenward title splash with the city's music swelling in the background. Painted Foothills (Dravanion Forelands theme) is one of my favourite area themes in the game.

For ARR, I'd say Coil's music is my favourite. I do like Gridania's theme. It feels like coming home.

It wouldn't surprise me if a lot of players do remember battle themes over area themes. Many areas you don't really have a reason to visit again after clearing the MSQ, especially if you don't engage in Gathering classes. Lots of players probably have heard the battle themes a lot over the years because of running roulettes for exp and tomes.

I do agree that the Yanxia theme is terrible though haha.