Elgato and Avermedia are basically the NVIDIA and AMD of the video capture hardware world. There's lot of generic "cheap" Made-In-Asia brands which might work perfectly fine ... or might not.
I would personally recommend an external (USB) model vs an internal (PCIe card) model. Easier to setup, easier move around between machines, always works even when the computer doesn't. The only real advantage of an internal model is that it doesn't occupy desk space or have any wire clutter. (Well that, and it doesn't get dropped as often, lol.)
Basically the important specs are signal type (HDMI, DisplayPort, etc) and resolution@frames (most are 1080@30fps, some up to 2160@60fps or more). Some of the external ones have storage capability (SD card slots, maybe even a SATA interface).
There is really no way to assign "unused" GPU outputs to video capture in Windows, it can be done with a linux hypervisor and KVM switching or GPU passthrough. There is really no way to avoid putting capture hardware in-line between your GPU and your fancy 165Hz gaming monitor.
There are also numerous video capture softwares. They always impose a performance hit, though it can be unnoticeable on beefy systems.
I actually use repurposed Android TV boxes for this task, lol, although they required some minor hardware modding and different firmware, but they're able to capture/decrypt/store streamed content, which I find handy.
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