I think it would be mostly your discretion on when you decide to divert. I don't really get your question, but I think what you mean is at what amount of fuel should you request a diversion and notify ATC. For this, it depends on the plane model, weather condition, and
For this example, let's say we're in a 737 approaching Toronto Pearson. We've just previously had two go arounds, and we'realmost on emergency fuel, with 40 minutes of fuel left. If the weather is clear, and you feel comfortable with the approach, you should probably notify ATC, but there is no reason to divert. On the other hand, if the weather is not clear and/or you don't feel comfortable with the approach, you can divert to another airport that you feel comfortable with or the weather is better there. , and if the weather's good, you can simply just declare an emergency and change runways.
If you divert, you should probably go to an airport that 1. is close to your current location (15 minutes away at tops) 2. has good weather and an easy approach, 3. meets your aircraft requirements. If you were at YYZ, the airports that you could divert to include Hamilton, Downsview, and if you've got enough fuel, you can try crossing the lake for Buffalo or Niagra. And if all 3 of these options are not able, you'll have to perform a ditching in the lake.
And as for the ten minute diversion call, it simply means when you have 10 minutes of fuel remaining + the emergency fuel. If you're at this stage, you MUST declare an emergency and divert if the weather or approach is not showing mercy.
I know this was a very complicated and long answer, but since we're all in quarantine, I figured we could all use up some more time. Hope I successfully answered your question, and I'm sorry no one else did.