My INTJ has a watch. He is very punctual. I think he may have reset the clock on my kitchen stove actually.... He also wants me to time everything I cook. My INTJ father in law is very punctual too. Hell, even the INTJ toddler keeps a pretty strict schedule-or more desire for a certain consistency. Things need to happen in a certain way. As long as those things happen, the rest is reasonably flexible. It's adorably OCD.
I don't think this is a type thing...at least for me..because i was brought up in a household that was very strict about discipline and social etiquette. Therefore i could say that my punctuality is a result that upbringing than my type. I have this horror of being late ..
INTJs seem to learn Fe "rules" as part of an NiTe system of social navigation when young? Even the INTJ toddler has already picked up very specific social mannerisms as memorized behavior patterns. Perhaps the strict adherence to time as a learned rule is part of this that some INTJs learn?
iN dominant with secondary eT creates the most arrogant of the sixteen types -the INTJ.
Hey ya cute little INTJ hater!

I'd suggest that the more strongly developed the eT to eS, the better the INTJ is at accomplishing the planned objective-molding the world to fit their vision, the less arrogant and obnoxious they become. (Or is this actually understanding how to form visions that are actually achievable? I dunno...) The ones i work with are older, can be forceful, but have a strong sense of honor and deep caring for those around them. They are not arrogant on the surface, although can inadvertently be dismissive at times. They are confident enough in their own capabilities that they do not need to make social displays of superiority or intellect, via insulting others around them. They are exceptional leaders-they act and others follow. They also have a fairly fast Ni reset time.
OP-Call her ass out on that behavior. Explain that she is being rude. Tell her that if she isnt interested in the relationship moving forward she needs to have the balls to say so, rather than play an avoidance game. If she really is interested, then it is something you need her to work on as it means a lot to you.