So there is no way to know if you are using the digital or the physical game itself...
Or check if the switch cartridge reader is faulty or not.
Also,for physical games switch games are not installed to the system right? Unlike PS4?
Unlike PS4, physical Switch games are not installed to the system.
When you place a cartridge in the cart slot on the system, the game icon should automatically show up on the left most window on the home menu. So, if you want to check if a switch cart is faulty or not, my best guess is to have your test cart and another game/demo, boot the non-test cart up so that game's icon moves to the left most window on the home menu. Close that game out, and then place in your cartridge. If the cartridge is working, I imagine that game's icon will automatically move to the left most window on the home menu screen.
As for how to know if the system is reading the cartridge, the SD Card, or the internal memory, my best guess is to count the loading time of whatever game. Digital Foundry did loading tests for the Switch version of Zelda: Breath of the Wild via internal memory, cartridge, and two different SD Cards, and the internal memory consistenly loaded the fastest:
https://www.youtube.com/watch?v=vbItTEmozxw
So if you own both a digital and physical copy of Zelda BoTW, and have the cart in the system, if your game boots from the title screen to the world of hyrule in roughly 30 seconds, then the Switch is likely reading the game from the internal memory instead of the cartridge.
Edit: I'll add that it's also been tested that Mario Kart 8 loads faster from the internal memory as well. I imagine this will probably be the case for almost any game, so you can do your own tests to see if the Switch is reading your cart or not for other titles.
