Hi L.,
While neither of my 2 children went down to one nap at just 9 months, I am glad that others here have told you that their children have. The reason I am responding is to tell you: don't pay attention to what books tell you! Our babies don't know how to read yet, so they don't know the "rules"! LOL!
My son (2.5 years old) started to give up his nap around 14 months old, but it was a 2-month process and you never knew from day-to-day what he was going to do. (I've read that several children can take that long to make the transition, while others do it much more quickly.) My daughter (who just turned a year old last week) started to go through the same thing at 11 months old. Still, I never know from day-to-day what her sleep schedule will be. It's a pain in the butt because it makes it harder to plan things, but I've just had to throw my hands in the air and so, "Oh well!" Kids never seem to do what you want them to do, WHEN you want them to do it!
I know exactly what you mean about your daughter not sleeping if there are things going on, but that she might need that 2nd nap if there is nothing going on. That is the exact same thing we are going through. Like you said, you are going to try this week to see if you can keep her up and just put her down for one nap--but if it doesn't work out, then just accept the fact that your daughter may need 2 naps some days and 1 nap other days. If she seems sleepy in the morning, put her down for a nap--if not, keep her up. (Heck, even if she seems sleepy in the morning, she might fight the nap! And that's okay too.) I know it won't be easy on you, but this is a temporary transitional period and before you know it, she will regularly be down to just one nap.
Sorry to make this longer, but I wanted to add something else: books and "experts" tell us how much sleep our children need, but those are just AVERAGES...they are not necessarily the amounts of sleep our own children need. My 2.5 year old son just started to transition from one nap to NO naps. Books say he needs at least 12-14 hours of sleep a day; he only needs 10. When he started to take himself off naps, I was concerned that he wasn't getting enough sleep, so I talked to a nurse at our doc's office. She told me that two of her four children never even came close to the amount of sleep she was told they needed; in fact, one of them AS A TODDLER would only sleep 6 hours a night, and had no naps during the day. He was fine and happy and fully functioning. So that just goes to show you that each child is different! You just need to figure things out through trial and error, and that goes for ALL aspects of raising a child, not just sleep issues.
(I just love Mamasource and other websites because I can always find someone else who has gone through the same thing I am when I find that my children are not acting as the books say they should!)
Good luck!
D.