What am I even looking for?!?

[KnottedYet]

More or less responsive.

If having a bike respond to your every command (intentional or not) feels wrong, you might say it feels twitchy or squirrelly.

My best example is with kayaks. I love the longish (17 foot) touring kayaks. I can't stand the chubby short puttering kayaks because they feel sluggish. Racing kayaks (long and very narrow) scare the ever lovin' cr@p out of me because they respond to every dang move I make and I seriously feel like I am gonna flip 'em over. Too squirrelly for me, but someone who loves racing kayaks is going to feel like my favorite touring kayak is sluggish and not responsive enough. It took a lot of kayak rentals for me to figure out what I liked best.

Ride lots of bikes, the more the better, and on lots of surfaces. You'll notice which bikes feel like they were made for you and which feel like you're engaged in a wrestling match.

Tire size makes a big difference, too, so also make notes of which tires you like on which surfaces during your test rides. I have no problem riding 23-25s on fresh gravel, but prefer 32s. So if I found a bike that felt marvelous in every way but wasn't quite up to snuff on gravel, I'd ask the shop to change out the tires before I gave up on the bike.