Good lord, man! I'm sorry, I'm sorry - I read this whole thing, along with all the responses it's gotten, and I'm sitting here shaking my head because you're making a BIG FRIGGIN' DEAL ABOUT ABSOLUTELY NOTHING.
Simple rules of the game, man:
1. You don't ask new girls out on weekend dates - Fridays and Saturdays are reserved for people actually IN established relationships. Asking for one of these days beforehand sends a subtle message that you view her already as THE ONE, but she's not there yet because you've only been out on THREE dates with her (which equates to, what, less than 6 hours of time together total??)
2. You always give a girl at least 2 times to flake. The first time could be a fluke and/or a real excuse; the second time means it's a pattern. She hasn't flaked on you enough times for you to start throwing a hissy-fit about it - when she said she had to look after the godson, it could have (a) been a real excuse, (b) a blow off 'cause she's losing interest, or (c) a way for her to avoid a Saturday date with you since, as stated in #1, her feelings for you are not THAT intense yet for her to claim you as the BF. In either case, YOU DON'T KNOW YET, so you give her the benefit of the doubt on the first one and go about your business like it's not bothering you - because it SHOULDN'T.
3. If she counter-offers, that means she's STILL INTERESTED. She said she couldn't do Saturday, but then came back with "How about Tuesday?" Um... THAT MEANS SHE STILL HAS AN INTEREST IN SEEING YOU. You playing this whole game of "Well, I'll just reject her counter-offer so I can control the frame and increase her interest" is stupid because she's already showing the interest by throwing out the counter-offer. Otherwise, she would have just said "can't see you tonight, I'll let you know when I'm available again" and never gotten back to you.
In short: you're over-reacting to this whole thing. Maybe it's due to fear of losing this girl's interest, or failing, or whatever, but it's not necessary. She said she was free Tuesday? Great - you could have totally taken it and not been seen as a beta who she can control (which I'm SO sick of hearing about on this site). And again, you have to remember: SHE'S NOT THE GIRLFRIEND YET. It takes at least 3 months for her to build up her interest to that level, so in the meantime only ask her out for weekday stuff, until SHE'S the one suggesting you do something on a weekend - that's when you'll know her feelings are becoming more concrete. Hope this helps!