> A computer can be connected to a network without having Internet access.
In fact, as the Internet is essentially a network of networks some of which may be connected and disconnected at arbitrary points in time, it is a bit difficult to strictly define what it means to have Internet access.
In practice it would probably mean something like "can access a large majority of servers that consider themselves to be Internet servers", which is however impractical to test for. A "good enough" substitute is often enough to test connectivity to a few representative services which are unlikely to be unreachable all at the same time if you consider yourself to have "Internet access" (Google, Amazon, ...).
In this case it would however be best to test connectivity between the user and the specific server that the user is supposed to access.