Generally speaking...

Shorter top-tube and longer seat-tube = more aggressive geometry?

Longer top-tube and shorter seat-tube = more relaxed?

You're on the right track, but there's more to it than that. I would say that the steeper the seat tube angle, the more "aggressive" the geometry is.

Longer/Shorter seat tube or top tube might just be a reflection of whether the rider has a long torso or long legs. In addition to the angles as mentioned, one thing you'll find indicative of a less aggressive frame design is a tall headtube. Aggressive bikes generally have short headtubes to allow as big as possible of a drop from the seat to bars.

Kind of, but it's more the relationship between the top tube and head tube length, rather than the seat tube. You can adjust for the seat tube length a huge amount by sliding the seatpost up and down, so the length hardly matters. But you can only adjust for head tube length with spacers and flipping the stem up, and you can only adjust for top tube length by longer and shorter stems, so you are more tightly locked into the dimensions the bike comes with in those two areas.

Seat tube angle plays a part in that the steeper angle allows shorter chainstays, reducing the wheelbase and making for a more nimble handling bike. But that's independent of the "aggressive" part which is more to do with the rider position.