My gut feeling is telling me that when James Phinney and Metzen worked together on SC, with Phinney probably writing most of the good stuff, Raynor was made into a regular soldier kind of a dude with regular accent and a balding head -
http://starcraft.wikia.com/wiki/File..._HeadAnim1.gif .
When he alone took over the lore, Metzen remodeled the character's look and added the heavy cowboy accent. I'll stipulate to SC1 Raynor simply having a "0" army haircut in SC1 and not going balding (though a careful look at the old portrait may indicate otherwise). Even sticking to the accent part of the issue, honestly, was there a code in the military that forbade one talking with the cowboy accent, which in turn prevented Raynor from speaking this way in SC1? Does drinking for four years make you start speaking like that? We can all guess answers to those questions.
It was more like...Metzen: "ha, I don't care what he looked and sounded like in SC1. I have personally always envisioned Raynor like this; say what...let's remake him as a cowboy, because I said so! Yeah!!!!"