An implementation by someone not involved in the original, and not using the original code, is not usually called a rewrite. The authors of this should use more careful language.
If the same author wrote one version and then another, and only a minority of the code (if any) is shared between the two, it's a rewrite. Especially if the new thing is intended to replace the old.
If one employee at an organization produced the first version of something, and another one at the same organization produces the second, using very little of the original code, that's also rewrite. The organization is doing a rewrite, even though that second employee is writing it for the first time.
Someone not connected to the original who knows only the interfaces, or inputs and outputs, isn't doing a rewrite; they are "cloning" or "implementing" and such.
If the same author wrote one version and then another, and only a minority of the code (if any) is shared between the two, it's a rewrite. Especially if the new thing is intended to replace the old.
If one employee at an organization produced the first version of something, and another one at the same organization produces the second, using very little of the original code, that's also rewrite. The organization is doing a rewrite, even though that second employee is writing it for the first time.
Someone not connected to the original who knows only the interfaces, or inputs and outputs, isn't doing a rewrite; they are "cloning" or "implementing" and such.