Confused about how to proceed

Your overall score is very good, and it will probably be perfectly fine for most of the schools you're looking at. I'm not an admissions consultant, so you'd do best to direct your question to someone expert about the programs that interest you, but the impression I have from test takers applying to those schools is that your Quant score is going to be a bit borderline for programs like INSEAD - it's definitely possible to get in with your score, but I think you could apply more confidently if the Q score was even a point or two higher. If your scores on GMATPrep tests were consistently higher than Q44, then you probably just had 'bad luck' on test day. Luck or variance always influences scores to some degree, since on some questions you will have guessed, and on some days those guesses work out well, and on some days they work out badly. If you were scoring, say, a consistent Q46 on GMATPrep tests, then if you took the test tomorrow, the most likely result is a Q46. On an unlucky day, you might get a Q44 and on a good day, a Q48, but it's not too likely you'll score at the lower end of that range. So if that's the situation you're in, I'd suggest just taking the test again very soon, and you'll very likely get a higher Quant score. The situation would be different if Q44 were around or above your diagnostic scores - then you'd likely need to make some genuine improvement in order to be confident of a better result on a retake. I'd add that you should only be using GMATPrep tests to gauge your level; company test scores won't be accurate enough to give you reliable information.

As for how your chances differ in Round 1 and Round 2, that's a question you should ask an admissions expert. Good luck!