[BUG] Cowork (Windows): project created with "Choose a folder" becomes a cloud project; local sessions then fail with misleading "Projects can't be included in sessions that run on this computer"

Environment

• Platform: Windows 11 (both machines)
• Plan: Team (same org)
• Working machine: Claude Desktop 2.9939.4 (MSIX)
• Affected machine: Claude Desktop, auto-updated, exact version not found in the UI

Summary
Two users in the same Team org created a Cowork project the same way on the same day, using "Choose a folder" to link a local folder.

• Machine A: the project header reads "Local projects are only available on this computer". Local sessions work.
• Machine B: "Choose a folder" was offered and used, but the project header reads "Created by you · Private", and the project behaves like a cloud (account-stored) project.

On machine B, every Cowork session in that project that uses the local folder fails with:

"Projects can't be included in sessions that run on this computer. Switch the session back to cloud, or remove the folder or permission setting that keeps it local."

Steps (machine B)

1. Create a new project in Cowork.
2. Use "Choose a folder" to link a local folder.
3. Open the project. The header shows "Created by you · Private", not "Local projects are only available on this computer".
4. Start a session in the project that works on the local folder. It fails with the error above.

Why this is a problem

1. The user explicitly linked a local folder, took the same steps as a colleague, and got a cloud project with no warning.
2. The error message suggests the session settings are wrong. The real cause is the type of project, which the message never mentions, so the user can't tell from the error that the project itself is the problem.
3. The only visible difference is the subtitle under the project name, which is easy to miss.

Expected

• Linking a local folder during project creation produces a local project, or the flow clearly says it will be a cloud project and why.
• If a cloud project is used in a local session, the error says so and ideally offers to create a local project instead.

Related

• #76694: "Choose a folder" missing when creating new projects after the Chat/Cowork merge
• #88632: Windows local sessions cannot attach projects (same error text)