You can actually have more than that. For example, the same private key can give you two legacy addresses - one compressed and one uncompressed.

The point about each private key only being linked to a single address applies for inside wallets. A single wallet will be all one type of address - all uncompressed legacy, or all native segwit, for example. Inside that wallet, each address will have its own private key. But it is also correct to say that you could extract a private key and use it to derive a different type of address if you wanted (although this carries additional security risks - much better to just create a whole new wallet of your desired address type.)