This fixes a regression from #33796.
I tried for several hours and cannot write a working test for this, but
this does fix the following warning in tests run with ASAN or TSAN:
-------- Running tests from test/functional/plugin/lsp_spec.lua
RUN T4667 LSP server_name specified start_client(), stop_client(): 114.00 ms OK
RUN T4668 LSP server_name specified stop_client() also works on client objects: 97.00 ms OK
RUN T4669 LSP server_name specified does not reuse an already-stopping client #33616: 31.00 ms OK
nvim took 2022 milliseconds to exit after last test
This indicates a likely problem with the test even if it passed!