[Regression]: "playwright.ps1 install" crashes in branches with older version when newer version was installed in another branch

Last Good Version

1.44.0

First Bad Version

1.45.0

Steps to reproduce

1. Create branch "A" where code references Microsoft.Playwright.NUnit 1.44.0

2. Run "playwright.ps1 install" - works as expected

3. Run tests - works as expected

4. Create branch "B" where code references Microsoft.Playwright.NUnit 1.45.0

5. Run "playwright.ps1 install" - works as expected

6. Run tests - works as expected

7. Change back to branch "A"

8. Run "playwright.ps1 install" - fails with error below

9. Run tests - fails with error below

ERROR:
node:internal/modules/cjs/loader:1148
throw err;
^
Error: Cannot find module './clock'

Expected behavior

"playwright.ps1 install" should not fail. If the versions are incompatible/different, then playwright.ps1 should install the version it expects.

Actual behavior

"playwright.ps1 install" fails

node:internal/modules/cjs/loader:1148
throw err;
^

Error: Cannot find module './clock'

Additional context

Cleaning the solution after changing branches is a valid workaround, but wasn't necessary before.

Environment

- Operating System: Microsoft Windows 10 Pro
- CPU: Intel64
- .NET Version (TFM): [net8.0]

Reproduce the issue with branches referencing Microsoft.Playwright.NUnit 1.44.0 and 1.45.0, using playwright.ps1 install and the listed Windows/.NET environment. Start by tracing how the script selects its version and handles the Node module error after switching branches. Done means playwright.ps1 install and the tests work again on both branches without requiring a solution clean.