Congratulations to your daughter for handling this herself, and handling it well! She knew she wanted only the one friend over, and she expressed to the second girl that while Daughter did indeed like seeing her, Daughter did not want see her THAT day. Saying "it's not a good day" is fine, as is "My mom says I can have one person over today" -- I don't mind if my kid "throws me under the bus" and makes me the bad guy in these situations!
I hope you do indeed have the second girl over, alone, the next day. Your daughter suggested it and should follow through, though of course if "the next day" isn't a good one, be sure you do have the second girl over at some specific time soon. Schedule it, so the girl knows your daughter meant what she said -- she does want to see this girl, but on her own. It is not "excluding people" to turn away someone who is trying to come into a plan that's already made, so don't worry about that. We sometimes tend to think it's excluding or being mean not to say yes to everyone all the time -- please don't let your child think that.
It's always OK for a kid to have time with just one friend. The second girl was being a bit pushy, but I would not put too much blame on her. She might not have a lot of experience with get-togethers or been talked with by her parents about why we don't invite ourselves on other people's outings. Not necessarily her fault there.
As for play dates or meet-ups with three kids: It's a good rule of thumb to have even numbers generally. Not all visits have to be just two kids; four is fine, if you know the kids' dynamics and know that they'll pair up if they get bored with each other. I have to say that as kids get older, you can forget the "it has to be even numbers" rule as well; my daughter is 13 and sometimes has two friends over. A great deal depends on the relationships among the three girls, and if you know your daughter's friends pretty well, you'll be fine with groups of three or more in a few years.
By the way, my daughter is an only child too. Not sure why you brought that up unless maybe you're worrying that only children won't navigate social things correctly or something like that--? Only kids are as outgoing or shy, as socially skilled or unskilled, as any kid with siblings can be. Your child stuck to what she had planned, didn't cave to another kid's persistence, and was polite in doing so; I think she's doing just fine!