Dk3
 

Can a batter, who swings/misses, legally run and acquire 1st base on a dropped third strike, when the pitch bounces over the plate or when F2 tries to "shorthop catch it" out of the dirt, but drops both examples? Or in other words, does the pitch have to hit F2's glove while in flight?

I've searched (see below) but would like rule reference please.

2-9-1 - Definition of a Catch..."secure possession in hand or glove of a live ball in flight", then farther down "NOTE: When a batted ball or pitch is involved, the above definition of catch applies".
2-28-2 Definition of a Pitch--nothing about a pitch that bounces
6-1-4 Legal Pitch -nothing about a pitch in the dirt
7-4-1b DK3 - "a 3rd strike not caught"
8-4-1e/f -Runner is Out - nothing that applies here

Caseplay 6.1.1c "A Pitch that hits the ground and bounces through the strikezone is a ball unless the batter strikes at it".

Caseplay 7.4.1.a/b Both give DK3 examples, neither with a pitch in the dirt

The proper terminology is not "dropped third strike," which makes it sound as if the pitch must hit the mitt in flight.

The proper expression is "third strike not caught." That should answer your question. You've got all the definitions you need: FED 2-24-2, 7-4-1; OBR 6.05c, 6.09b

What more do you want?

Yes. No.
This has been another case of simple answers for simple questions :)