How do you know when to stretch? What is the "push notification" system the body has for that?

Y'all missing the point. OP isn't asking when to stretch. The question is what kicks off the feeling. Pretty certain it has to do with blood flow, as that's the point of stretching.

When your muscles feel tight or tense, or a bit achy, you'll automatically feel like stretching them.

If my joints feel a little stiff, or if I bend a certain way and it's not real comfortable, then I focus on that and give it an extended focused stretch.

Also, it's a good idea to stretch before any physical activity.

Every day. When I don’t, the odds of injury goes way up. The push notification is pulling a muscle.

Muscles and joints have receptors to feel the stretch. If you are not in a comfortable position since a long time (according to the receptors), you want to stretch.

You don’t normally, until you get old and realize you’ve lost a ton of mobility. It’s something you need to actively notice.

usually when I have to poo.