Yes and no. In principle you are right.

In practice, Claude is trained on its harness and the subscription is priced to best competitors such as Cursor.

This is also why Cursor tries to finetune oss models. Otherwise its performance in the CC flavor of AI coding will just be that bit worse

If you install the VS Code plugin, it's the same editing functionality. Cursor lacks a lot of the tooling in claude code that makes the experience a lot more... solid.

That is my experience currently.