I've become a really good putter. Above average for my handicap in both percentage of one-putts as well as 3-putt avoidance. IMO, there are three things you have to learn to do to be able to do well in these situations.
1. Learn to read situations really well.
This is more than green reading, though that is a big part of it. It's also knowing the implications of different misses.
Example: A breaking putt that falls away from the hole can easily get away from you. Putts like that you have to die around the hole. Or maybe it's flat on one side of the hole, but drops off steeply on the other side. You must keep it on the flat side of the hole in the scenario you mentioned.
2. Develop great distance control.
Inconsistent setup, inconsistent contact, inconsistent swings, inconsistent tempo, decelerating, using fast twitch muscles to power the swing. All this has got to go. The best golf tips I ever got were these two putting tips. Power the putter with the big muscles and straight back and straight through. Obviously, this requires lots of practice to groove and maintain. About half of your practice putts should be lag putts outside of 25 feet and from all different holes and angles.
3. Become a great putter inside of 5 feet.
About half your practice putts should be 4-8 feet out with all kinds of different breaks and green speeds. Play lots of games that make these pressure putts.
I learned a lot about my tendencies through this process. Both the stuff that gets me off track and the keys for me to make a lot of putts. But you have to pay attention and practice a lot to reap those benefits.