You are correct in that an IRI is an instrument rating instructor, whilst the IMC rating can be taught by someone with the 'no applied instruments' restriction removed from their FI rating. I used the phrase IRI incorrectly, but it was to save a whole bunch of typing.
I'm an IRI who also teaches IMC and I don't teach them differently, apart from a a bit of tolerance, but I like everyone to be as good as possible given the limitations of private flying.