The new Handler calls h.ServeHTTP to handle each request, but if a call runs for longer than its time limit, the handler responds with a 503 Service Unavailable error and the given message in its body. (If msg is empty, a suitable default message will be sent.) After such a timeout, writes by h to its ResponseWriter will return ErrHandlerTimeout.
-- https://pkg.go.dev/net/http#TimeoutHandler

For incoming server requests, the context is canceled when the client's connection closes, the request is canceled (with HTTP/2), or when the ServeHTTP method returns.
-- https://pkg.go.dev/net/http#Request.Context

If a call runs for less than its time limit, but the client cancels the request, TimeoutHandler will respond the same way -- with an ErrHandlerTimeout.

This makes it hard to tell if the code is performing poorly and hitting a timeout, or if the clients are ending requests.

I propose a conditional is added here, to check if the context's error is a context.DeadlineExceeded or context.Canceled, and return different errors.