I think the reason for the semantic ambiguity is that there are several forms that are getting mixed up
1) nir + mā (nirmāti or nirmimite p.p. nirmita) whose primary meaning is build or create
2) ni + mā (nimimite, p.p. nimita) whose primary meaning is measure
The Pali for #2 is nimināti ("measures, exchanges, barters") p.p. niminita
The Pali form for #1 is nimmiṇāti ("creates, fashions, builds"), p.p. nimmita.
Nobody knows where nimitta comes from. It looks like a past participle, but of what verb? Skt also has a verb mi which means to "fix, fasten in the earth, set up, found build", p. p. mita ("set up, established, etc."). The verb also appears to have an alternate form mit per MW. ni + mi means "to fix or dig in, erect, raise" and is attested in the RV. It also means to "perceive, notice, understand", attested in the AV. So that would be a good candidate for the origin of nimitta, if ni + mit was used in historical times: ni + mit + ta, but it is not attested as far as I know.
Then there is the verb you mention, mid which has the meaning "grow fat" and also goes back to the RV. But nothing is attested with the ni suffix, that I have been able to find. This verb is also related to mith (see MW sv. mitra) with meaning "unite, pair, couple", but again, there is no ni + mith, that I can find.
In Pāli the only forms we have are nimitta and nimmita ("created, fashioned").
We know that in the first manuscripts written down in the first century BCE, geminates were not written down (Norman, Philological Approach to Buddhism, 107) . So the word would have been written as nimita. It was up to future scribes, when transcribing back into "correct" Pāli to determine whether the -m- or -t- were geminates or not. Now if there was still a bhāṇaka around and he pronounced the words with the geminates intact, then the scribe might have had a chance of getting it right, but if not, it was up to the scribe to choose. Exactly why he/she chose nimitta, I don't know (Skt. nimitta is fairly late, from MBh time), but it looks like some of the meanings of the other verbs got mixed up so that nimitta is a composite of ni + mā (measure), nir + mā (build) and perhaps ni + mi(t) (erect, raise, perceive, understand) and perhaps ni + mid ("meaning"?) that you mention. I don't know, but it looks like nimitta arose because of mixing up all these verbs.
Anyways, these are just some thoughts. I don't know whether we'll ever understand the derivation of nimitta, but we can speculate. If it was formed regularly, then it would have to come from ni + mid + ta as you suggest, or ni + mith + ta, which would both produce nimitta in Prakrit. where did you find the derivation for ni + mid + kta with the meaning of "cause" and "sign"?
Best wishes, Bryan