Can websites, apps, games etc. tell that I used MitM on my own computer to view their HTTPS traffic and then re-encrypt it and send it to their servers?
Server applications do not distinguish clients. MitM is no different from server application perspective than legitimate client.
Only client can detect (under certain circumstances) that its traffic with server is listened by someone else. If MitM is not legit, an application will present error messages that server certificate (which is actually MitM cert) is not valid/trusted. This is an indication that someone attempts to listen the traffic.
In enterprises, it is hard to detect when company implements a HTTPS inspection on their gateways, because company has control over all enterprise applications/devices and can install CA cert used by HTTPS inspector into trust store. In this case, MitM will be transparent and undetectable.
As the last resort, client applications may implement HTTP Public Key Pinning which will detect the most sophisticated MitM scenarios, however this approach is not very popular because of expensive maintenance and additional implementation in the app code.