You've expressed it better than I did...
Here's my take on it.
You tap out to acknowledge your inability to deal with the technique. You don't need to feel pain to tap out. When I first started judo and bjj I wouldn't tap until I felt pain or was choking badly. I'd go home with sore arms and sore throats.
Now I understand better that tapping is about admitting helplessness. I can tap the moment I know I'm not escaping. In terms of bjj/judo training. I'll tap when the technique is locked in and my attempts at escape have failed. In kata terms I'll tap when the technique is correctly applied and the only point in continuing is to cause pain/injury.
Being able to 'take' it has no bearing on when to tap. Pushing limits in pain really doesn't have any purpose in knowing when to tap.
The number one rule is "protect yourself at all times". It is your job to know your limits and to express them. It is your partner's job to respect those limits. When you feel in danger, or helpless, or lost, it is the time to tap. Only that will prevent injury.