Currently a Utah resident and have been for 25 years. (don't judge me :-P)
But I can confirm that I catch myself frequently stating "mountain" as "moun'ain", Layton as "lay'on".
However, I have never vocalized "creek" as "crick" (though it seems to be about 50% of those I've come across) or "Mormon" as "Marmon".
I personally would say that the Utah accent is distinct because it's similar but not completely 'country' mixed with an almost southern twist.
As far as different areas of Utah having their own play on the accent goes: yes, in Provo I've noticed a fair amount of 'better' annunciation of words like mountain where the 't' is actually heard (though still faint and abrupt) along with habits of emphasizing words in the sentence, as a whole, differently then what I would expect to hear.
Example- in Provo (generalized and subjective) any sentence that begins with "I" where the actual emphasis is placed on a verb, "I like runnin'!", tends to be pronounced with a double emphasis, "I like runnin'!", and to me sounds terrible.
In opposition, hearing this same sentence in West Valley sounds more appropriate as there doesn't seem to be the initial emphasis on the beginning "I".
As a side note, a very prominent part of the Utah speech pattern is the use of spin-off phrases. "I'm hella excited!" turns into "I'm hecka excited!" and not just for the obvious reasons. I've found that most people I've come across who have different religious beliefs than the majority of the state, still have adopted 'hecka' as a preference to 'hella', though the same person will have no inhibitions to dropping 'f' bombs.
Then there's the 'uh' sound in most words. You don't here, "You all take care now!", or "Ya'll take care now!", but instead, "Y'uh all take care now!".
And finally the other speech pattern that I've noticed to be very common here is the contractile contraction. "It's all fun and games until someone gets hurt." turns into " 'S'all fun'n games s'til' someone gets'hurt."