I just added
# set PATH so it includes user's private bin if it exists
if [ -d "$HOME/bin" ] ; then
to the .bashrc script, to get the behavior I want.
Apparently Gnome will carry through the setting of your initial login in all subsequent bash sessions while LXDE does not (or maybe some other component that is different from Mint Debian Edition and Sparky, there are probably only a few billion differences, I do not claim I understand). At any rate .bashrc is what is supposed to, and does, execute for non-login shell and when I insert the above into it I get the behavior I expect.