What you say about partially updated flag registers is right.

Because of this, most modern ISAs take care so that the flags register is always updated completely.

In legacy ISAs, like in x86-64 where the carry flag is updated or not updated separately from the other flags, it is handled by the CPU as a distinct register, so the carry flag is renamed independently of the other flags.

Moreover, in x86-64 the overflow flag can be used as second carry flag in some instructions. Having 2 carry flags permits the elimination of some functional dependencies between instructions that would not be eliminated by the renaming of a single carry flag (renaming solves only resource dependencies, not data dependencies).