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...