Should the invitation be worded so the groom's parents are credited as being the hosts? or should the invite simply say something like, "You are cordially invited to a rehearsal dinner"......and so on.

I think the question you need to answer first is whether you want to highlight or obscure that they are hosting. If the issue is that they aren't paying 100% or something, I think you can ignore that -- it's none of the invitees' business. (Whether the other people paying care is an entirely separate issue!)

If you want to sort of obscure the fact that the groom's parents are hosting, the invitation could be something like this:

Rehearsal dinner

Date
time
place

RSVP info [which might give a clue, if you include a name or an e-mail address]

But if you want to show that they are hosting, you could either do it as lindac suggested, or, if you want it informal, something like this:

Please join us for dinner [or "cocktails and dinner," or whatever]
date
time [or "immediately following the rehearsal," if appropriate]
place

- Mary and John Groomsparents
RSVP

The rehearsal dinner invitation will be worded so that the groom's parents are shown to be the hosts.

The invitation for the party after the dinner ( all out of town guests are invited) will simply be an invitation to a "meet and greet" party so family and friends of both families can get to meet each other at an informal affair before the big day.
No hosts will be specified for this function. We didn't feel it was anybody's business to know who was paying for this party and we're okay with this.

I'm looking forward to a wonderful wedding weekend and am pleased we could come to a compromise we could live with.

Thanks again for your sage advice.