Bella is Edward's girlfriend in private, but she dislikes acknowledging this in public. Why? She is not a vampire and Edward is...having this become public knowledge is damaging to their relationship to the point that they would be forced to remain apart. The vampire society is a secret one, and cannot be illuminated. Bella considers them to be "good" vampires since they have pledged to survive on animal blood and not the blood of humans; however, it is important to recognize that Edward, in spite of this, considers the vampires doomed as a result of their dark lifestyle.
Edward's vampire friends knows about the relationship, but Edward encourages Bella to keep their status secret from the human community. His motto? Better to meet and love in secret for as long as possible...forever, perhaps...than to be forced apart by two communities who don't understand their passion. "Fiancee" puts a target on her back, so they choose not to use it or any other term of endearment when they don't consider themselves safe to freely express themselves. Sounds a little like Romeo and Juliet, doesn't it?
She thinks that if she is marrying him right out of high school it is a sappy romance type thing small town girls do and she feels that marriage puts a deadline on things. She believes her love is stronger than marriage, for all eternity, which is also why instead of growing old with Edward it was more like staying by Edward forever and all of eternity.
Although Bella can come off as a hopeless romantic something's she says can either be seen as very deep or something along the lines of a clichéd hopeless romantic. The reason Bella does not like the be called Edward's fiancée is because she feels that marriage is something that puts an expiration date on their relationship and the thought of marriage is not as strong for the feelings they have for one another. Basically for Bella her love for Edward is stronger than the label as a fiancée.
I thought the reason that Bella disliked being called finance was because she was initially oppposed to marriage itself. 'Finace' is a label that relates to marriage. However, I do agree with Edwards point of view in keeping their human/vampire relationship low key.