The SHIELD name had been in use long before the 90's, but it wasn't well-known publicly before the Avengers stuff, and that makes all the difference.
In Winter Soldier, computer-Armin Zola tells Captain America and Black Widow that SHIELD was founded and had recruited him right after WWII - he doesn't use the full words, he actually says "SHIELD". Although it could have come later, it implies the name was used even back then, as it's shown on his security clearance document. Zola transferred his brain into the computer in the 70's, so at the least, it would mean the Shield name would have to have been in use before that (again, it's written on Zola's identity document). But Zola was an "in". He was recruited by SHIELD and would know what it stands for, so he uses the shortened name.
Coulson uses the full words when he introduces himself to Pepper because he is assuming that she has never heard of the organization, much less by its acronym SHIELD - and she actually hadn't at that time, so he's right. She says they should shorten it, and Coulson says they are working on it, though he knows they already have a shortened version, he's just polite enough not to dismiss her suggestion. Plus, at this point, she is still an untested ally. By the end of the movie, she's an "in," too - she's worked with the agency by that point and Coulson expects that they will work together again, so when she starts going into the full words of the agency, he lets her in to just calling it SHIELD.
Fury not using the full words is probably more just a personality thing. He might assume that Captain Marvel is an "in" because of what he was briefed on before they arrived to meet her (or maybe thinks she would know what SHIELD is based on her actions), but it's probably more that he just doesn't have the politeness that Coulson has (i.e. Fury doesn't care if she knows what SHIELD is or not), and instead prefers the time-saving use of just "SHIELD". Introducing himself that way also puts Fury in a power position - if someone doesn't know what SHIELD is, then there is the conclusion that Fury either has more knowledge or more clearance - and Fury would happily put himself in either of those categories.
So ultimately, long explanation, but it's probably just that Coulson and Fury introduce themselves differently from each other, or differently depending on who they are talking to.