> they're identically-named things that both repackage OpenAI's LLMs

Haven't tried it yet but the GitHub Copilot extension for VSCode also seems to integrate Claude, Gemini and other non-OAI stuff

They do, and those models are served by Microsoft. You pay a premium per “request” (what that means is not fully clear to me) for certain models. If you use the native chat extension in VSCode for GitHub CoPilot, with Opus model selected, you are not paying Anthropic. This counts against your GitHub Copilot subscription.

The Claude Code extension for VSCode from Anthropic will use your Claude subscription. But honestly it’s not very good - I use it but only to “open in terminal” (this adds some small quality of life features like awareness it’s in VSC so it opens files in the editor pane next to it).