The government (in the form of the SIA) has been wrestling with the definition and still isn’t sure. You see they are trying to license Private Investigators in the UK and in order to do that they have to define what they do.
In all their definitions, what is clear is that a Process Server is not one. As far as I’m aware, a Process Server will not need a SIA license.
Often a Private investigator is also a process server. The main difference is that a PI will be able to carryout investigations and inquiries while a process server will not. They will only be able to serve process. No house to house inquiries, no tracing if the person is not at the address specified.