I've never seen 'puter with an apostrophe, but puter is common enough among geeks, though always playfully, never as the standard term. It's just a lopping of the first syllable, not a slurring.

Now I've just consulted the Jargon File (a very good and comprehensive survey of geek terms) and it's not in there, so I'll have to guess. There is (was) a thing called a transputer, which is a kind of chip with a certain innovative architecture. My understanding was that puter was originally either a synonym of this or a generalization into a morpheme: so that a standard computer and a transputer were two different kinds of puter. But as the transputer is now dead, that generalization must have died with it.

My girlfriend says puter to me and putie to her children.