How to Control Which Apps Show In Your Desktop Environment
I'd like to make it so when I use Trisquel that any Gnome desktop apps don't appear when I run them through Synapse and vice-versa.
For example, if I run Gnome Desktop, I find that in some cases Caja shows up instead of Gnome's file manager. Is there a way to control this?
I'm totally guessing, but maybe you're missing a GNOME metapackage that tells the system which apps to install and use by default with GNOME?
I'm not 100% sure I understand what you're asking, but applications that appear in menus have a .desktop file in /usr/share/applications. Desktop-specific applications typically have an "OnlyShowIn" line in their .desktop file that specifies which desktop environment(s) should show that program.
For example, /usr/share/applications/caja.desktop contains "OnlyShowIn=MATE;", so it should show up in a MATE menu, but not in GNOME's dash. However, being a desktop-agnostic launcher, it is also possible that Synapse does not respect this. It might not have a way of knowing what desktop environment it is running in.
If you are running Etiona, you try installing mate-applet-brisk-menu and add Brisk Menu to the botton panel in place of the default. It is a nicer menu in my opinion, and it has a search bar for looking up applications like Synapse does. Being intended for MATE, it should avoid showing applications only meant to be shown in other desktop environments.