On the other hand, using the fact that limen can mean the upper part of a door frame is no argument for using threshold as a word for a lower limit that must be gotten under to proceed.

I made no such argument, and I'm sorry I expressed myself so poorly that you believed that I did. That being said, I have no problem with using threshold to mean 'lower limit'. I see a threshold as being a threshold no matter whether you enter the house over it or exit the house over it. I see it as a sort of terminus both ante quem and post quem.

Ceci n'est pas un seing.