Sounds like you were doing it right.
Not sure about your questions. How could the fitness tracker take those 10 minutes into account if you weren't tracking them, only the 50?
A "relaxing savasana" doesn't sound like "exercise". Is that where you lay down and they talk you through relaxing each part of your body? That isn't exercise and I wouldn't track it. The warmup is more variable. If you feel it's getting there, it's okay to track, but as what, exactly? Since you're sitting I would say the extra burn in the warmup is quite marginal and probably also not worth worrying about.
Remember that yoga itself is not a calorie burner. You do not need to worry if you're counting your calories right for yoga - as you shouldn't be counting them at all. I would use the Quick Tracker entry and enter 55 or 60 minutes as recording your fitness time spent on your flexibiilty, but not use the cardio tracker (unless it's the power yoga that is listed there, but personally I don't even consider that to be 'cardio').
Deb, in New Zealand