AdMech characters start with this. Now, is a vox-caster a megaphone (equivalent to a Laud Hailer) or does it include a 'radio' (what we would understand as a 'vox') as well?

Or the opposite of a radio, a long range transmitter. Good question.

Page 203, OW CRB:

Going by the description of it, and its depiction on miniatures, it appears to be a radio telephone, the sort a forward observer for artillery might lug around to coordinate with artillery batteries for example.

For the Tech-Priest, it's probably just an integrated radio transmitter/receiver.

Those Vox-casters carried by IG troops, though, integrate loudspeakers so that orders received from command can be broadcast en masse to the troops around them, correct?

In the OW Laud Hailer is a separate item (Corebook, p.201).

I would say that it has radio function.