Use tsd for type testing

Issues #4160 (PR #4366) and #4202 involve changing some of the declared types in the API. To minimize the risk of regressions (i.e. type errors in code that previously did not produce them), it would be helpful to explicitly test API types. I'm testing out tsd, and it seems to work quite well, but unfortunately I ran into dependency conflicts, and someone more familiar with the codebase would probably be in a better position to resolve them.