First warning you got seems suspicious (found more than one fuel-core ..) so let’s try to remove non-fuelup fuel-core binary in your path first. It is highly possible that you might have installed it via cargo, or built it from scratch so I would suggest taking a look at ~/.cargo/bin/ and try to see if you have a fuel-core binary in that folder, delete it if there is one.
My fresh installation succeeds to install fuel-core with fuelup latest. So it is highly possible that you have another fuel-core installation hanging around in your path. I would suggest trying a clean installation and keep on using a single installation method. This is pretty important as this is an inherent problem with modern operating systems.
Alternative solution/suggestion
Each “conventional” package manager takes up a place in your PATH order, and it is not possible to deterministically re-create environments such that they do not coincide with each other. So using multiple installation methods requires users to be aware of which binary takes the presedence etc. Easiest approach there to stick with only a single installation method.
Another solution would be Nix, which is an open effort that tries to solve package management problem in a declarative manner. That makes sure that you got the developer environment exactly as intended. Fuel also maintains a fuel.nix flake if you want to give it a try. Bear in mind that it is not completely ready to be used with m1/m2 macs, which would lead long build times. We are working on it to get m1/m2 binaries ready to be used. The documentation book is getting ready atm but you can read the quick start from here.