I always considered the service to be what the POCO supplies, that the premises wiring and utilization equipment get their power from.
As in "POCO provided a 200A service, at a cost of $XX".
The service point is the arbitrarily designated (by POCO and maybe AHJ) point where POCO responsibility stops and the customer's starts, even though POCO may also still impose conditions on the wiring downstream of the service point. The service point is also generally considered to be the designated point where NESC (or other POCO adopted standard) jurisdiction ends and NEC jurisdiction starts.
The service point is at or somewhere upstream of the service disconnect (which is fully defined in the NEC).
The NEC puts limits on how many services there can be to a building, but more than one set of service conductors from the secondary of a single POCO transformer could be either a single service or multiple services. The NEC does not specify how that distinction is made.
Two sets of service conductors (not paralleled) from two different POCO transformers/transformer-sets would seem to me to always be two services, but....