You are talking about the person's SUN sign.
That is the factor you identify when you look up their month-day on a list.
But Sun-sign astrology isn't really astrology. Sun signs are the only factor, out of 40-50 factors, that can be easily found, without extensive calculations.
So it is easy to sell and make money off it.
But it isn't really astrology.
A person's Sun sign indicates how they relate to themselves.
Romantic/sexual interest, and behaviors arising from such are NOT indicated by the Sun's position.
You want to look at the individual person's Moon (emotions), their Venus (relating style, relationships needs), their 5th House (specifically about romantic style), and their Mars (how they take action to get anything that they may want).
But you don't need astrology to answer your question.
IF he is relationship material (meaning, among other things . .not too shy) and IF he is interested in you . .then he asks you out and doesn't waste time asking your out (some other guy might beat him to it).
If a guy is not asking you out shortly after he first meets you, then either you do not attract him and aren't going to attract him, or he is too shy to start a relationship (which also means he lacks the emotional courage to handle the problems which always start up sooner or later .. .so you won't succeed with him).
From what you say, it sounds like you don't have a chance with either of them.