I'm going to disagree with some of the advice posted here and suggest that there is a lot more going on here than simple petty jealousy.
Why was your boyfriend ambivalent about ring shopping? Is he just ambivalent about the ring (outdated symbolism, etc), or are you detecting ambivalence about the marriage itself?
Why was he so interested in texting his girlfriend during the ring shopping, instead of being present in that special moment with you?
In the other texting/phone incident: you had fears that his phone hadn't really died and that he kept texting the girlfriend while indicating to you that his phone had died. Are those fears all in your head, or valid in some way?
I think you and your fiance should have some serious discussions about these issues. It might be resolved with better communication; or it might be a sign of some deeper problems.
The way you describe your fiance's relationship with the other girl is somewhat vague. Is she his girlfriend? Do they use the label secondary? Or is their relationship more casual? Are they committed to a future together in some way? How do you feel about their relationship?
As far as the original incident with the girl(friend) running over to greet him in the restaurant: maybe she is just more relaxed about public affection / showing excitement in public. I can see how you are tempted to interpret it as her not respecting your relationship with your fiance, but maybe she's just the kind of person who runs squealing to greet all her friends.
My lover has a female friend like that. They are very physically affectionate in a platonic way; they're not involved sexually and don't want to be (although he is free to date whomever he wants). She always runs squealing to hug him when she sees him. I found witnessing that very awkward at first, since I don't express excitement or affection that way. Now I'm used to it and it's no big deal.
However, Coraline, your post reminded me of times in past relationships where I experienced feelings I dismissed as petty jealousy / my own problem to deal with. In fact, what I was feeling was a tremor of warning at a red flag. I should have been paying attention to the red flags instead of saying, "This is my pettiness to deal with."
Example: My ex and I went on a week-long trip to my childhood vacation home in the mountains, which I had put a lot of effort into planning. Neither of us had ever spent a whole week with any romantic partner before. I had never before brought a boyfriend to this place that was so special to me. My ex spent a lot of the time glued to his phone/device, which irritated me, but I told myself that he needed to be "plugged in" for all his business stuff and that it's so cute that he's such a techie geek that we're surrounded by beautiful mountains but he needs to look at his email.
I should have noticed that he wasn't "present" with me or connecting to me emotionally in any way. That was one of the signs that I misread.
I'm not saying you definitely have red flags here, but it's worth considering.
Single, straight, female, solo, non-monogamous.