esp8266 connect when starting to which AP?

Regarding caring, it is about the ESP connecting to a weak AP when there is a stronger one available...

Why do you care? Suppose that you have two pitchers. One can throw a fastball at 130MPH. The other can throw a fastball at 100MPH. You've got a batter coming to the plate that has never been able to hit a ball traveling more that 10 MPH. Does it matter who pitches to the batter?

If the ESP connects to an access point that is good enough to get data to and from a server, why does it matter that there is another access point that might, at the moment, be better?

What would be better would be for you to take your ESPxx to the mysterious location, and see what it actually does.