Can I run Bitcoin-Client and Bitcoin-Full-Node on two different computers?

Bitcoin Core software only implements fully validating nodes. There are no other operating modes, so it always downloads all blocks, verifies them, and optionally saves them. When running in cleanup mode, only the last few 100 blocks are kept on disk, but all previous blocks are still downloaded and fully verified.

However, there are many other Bitcoin clients, many of which are much more lightweight. Some of them can be connected to specific nodes of your choice. However, software recommendations are off topic on this site.