First - do you understand the terminology? Presets are the various parts of a complete Rig. An AMP preset is just the amp, a CAB preset is just the CAB, any effect preset is only that effect and so on. Anything beneath the All Presets section are these component parts.
Your screen shots all show a Player, not a Stage. For other profiles, it would normally say My Profiler where your screenshots show Player.
So, if you wanted to save just the AMP part of your rigs so they show up on that dropdown menu, open the rig you want in Rig Manager. Drag the AMP (the Amplifier icon in Rig Manager to the Player section that is beneath All Presets. Same thing applies if you wanted to save CABS, Drives, Delays, Reverbs, etc. If you drag them under Local Library, you would be storing the preset to use later.
Also note that under the Local Library sections, you can make your own folders and sub-folders to organize as you'd like. You can't make folders under other sections.
Player (or My Profiler) is your hardware device. Local Library is for storing things on your computer but not currently loaded on your Player. Anything under the All Rigs section are full rigs. Under Player means on your device. Under Local Library is stored and available to put onto your Player.
All Rigs are full rigs. All Presets are the various parts of rigs.
Hope that helps.