A grenade has range Short if thrown by hand or shot by a grenade launcher?

Thrown by hand would be my guess. Using a launcher, I'd peg them at Long Range, but the launcher would require the Gunnery skill.

The underbarrel grenade launcher in the attachments section seems to be for special micro-grenades, which are a good deal weaker than the average frag grenade listed earlier in the weapons chapter.

Sorry I was talking about the Grenade Launcher mounted on a AT-PT.

It's a personal scale weapon that has Short range, but is seemed too short to me…

Since EotE breaks the scale up into "character" and "vehicle/starship", the AT-PT was probably written using vehicle/starship range bands, as discussed on page 158 of the Beta.

Short version, "close" and "short" range for vehicles is a heck of a lot further than "extreme" range for character-scale weaponry.

I know the AT-PT has a disclaimer of using personal scale and not vehicle scale, but I'm thinking that's more for the damage than range.