Trying to read from your hdmi connected TV doesn't work this way. Data being sent to the TV by one device (your laptop) cannot be read by a different device (your mini pc) also connected to the TV by hdmi. The hdmi cable is just an output. Data can only flow from the PCs to the TV, not from one PC to the TV, and then from the TV to the other PC.
The easiest thing you can do is to use just one PC that does it all, so OBS can capture everything on that machine.
If you must use a separate PC for your TV, you need a hdmi capture device with a hdmi passthrough. Such a usb connected capture device is able to read a hdmi output and provides this like a webcam to a PC. Connect that capture device to your mini pc with USB, and connect the hdmi output of the laptop to the hdmi input of the capture device. This way the hdmi image created by the laptop is available as webcam device in the mini pc. To still have the image on the TV, connect the hdmi passthrough output of the capture device to the TV.
That hdmi passthrough output is optional. If it's sufficient to see the TV video just within OBS on the mini pc, you can go for a cheap capture device without passthrough.