I read a little bit in there, but I don't understand much of the subject, I know that my laptop only has an HDMI output and I need an input, the Internet says that I need a capture card, but there are different ones, some are 20-30 euro and others cost around 200 euro, an exact example would be good for me.
The cards from AVer Media are cheap and good, have the LGX2 yourself.
Otherwise an Elgato HD60S is also recommended.
Only works with an Elgato
The answer depends on what you want to do with the incoming video signal.
Just so I can use my laptop screen to play
Then forget it - you won't be happy because of the topic of latency.
If it is e.g. B. For image viewing and / or video, you could use something like https://www.airserver.com/ if you could get AirPlay, MiraCast or Google Cast to work on the switch. I don't know if it supports something like that…
If you have enough electronics know-how, then it would of course be conceivable that you tinker an HDMI connection to the monitor cable of the laptop. But then you could moth up the rest of the laptop.