An injured household employee is a different claim
If someone you employ directly is hurt while working at your home, that is an employment injury rather than an ordinary guest injury. Personal liability may not respond to it at all, and the correct cover is workers compensation, which many households do not realize they can or should carry.
Employee or contractor is a test, not a label
A pool company sending a technician is a contractor with its own insurance. A housekeeper you schedule, direct and pay directly looks much more like an employee. The distinction turns on control rather than on what either party calls it, and it determines who answers when something goes wrong.
Some umbrellas extend to employment practices
Claims by household staff alleging wrongful termination or similar are excluded from most personal policies. A few umbrellas offer limited employment practices cover for domestic employees. Where staff are employed directly it is worth asking for by name.