I have a history of kidney stones, and what you are describing definitely sounds like a UTI and not like a kidney stone. Usually with a kidney stone, it's all about the back pain and not about pain associated with urination. I say usually simply because that has not been my experience. Also, I've talked to many other people who have had kidney stones, and their experiences have been very similar to mine. Still, I'm not an expert.
I have had UTI's before, and what you're describing sounds about right. Even though what you are experiencing is tolerable, in the future I would not wait over a weekend for the doctor's office to open. If you have an urgent care nearby, I would go. The thing about UTI's is that if they are not treated soon enough, they can develop into a kidney infection. I had one of those about a year and a half ago. It kept me in the hospital for 5 days, followed by 3 days in bed at home. Kidney infections can be very serious, so this isn't something you want to take lightly.
If you can go to an Urgent Care or Prompt Care tonight, do it. It's important to get this checked out as soon as possible.