How can I make a program run at startup in only one definite session that uses the same desktop environment as others?
Solution 1:
Startup applications are defined by .desktop files that live in ~/.local/share/autostart, according to the Desktop Application Autostart Specification.
They optional setting to limit them to one or more desktop environments, e.g.
OnlyShowIn=GNOME;XFCE;LXDE;
Alternatively you can use
NotShowIn=...
to exclude this app from the listed desktop environments.
Only one of these keys, either OnlyShowIn or NotShowIn, may appear in a single .desktop file. (That was a direct quote from the spec.)
Unfortunately, I'm not sure what you're doing qualifies as a separate desktop environment...