I'm thinking of introducing my significant other using those words
I don't think that would be a good idea
I'm afraid that I may not be able to introduce her as my girlfriend
It's okay to not want to be too presumptuous in introducing your girlfriend as such to others. However, she might be wanting and expecting you to acknowledge your relationship, so maybe you should see how she feels before you use such bland terminology.
It's okay. I'm just going to call her my fiance since we have been in a relationship for so long already.
I really don't think you should be introducing her as your fiancee unless you are already engaged to be married. Calling someone a fiancee doesn't just mean that you have been dating for a long time, it means that you have proposed and she has agreed. If you don't know how to introduce your significant other, I strongly suggest that you ask her what she is comfortable with.