[So when all this happens, IS THIS A DIAL UP MODEM CONNECTION??]

maybe it's less confusing to call it this - DIAL UP CONNECTION
that's what it would be.

I think they call it "dial up" connection as opposed to DSL connection or cable connection etc, and because it has to dial the ISP service every time you connect using the telephone line.

She does understand will not have use of her phone for incoming or outgoing calls when she is online using the dial up.
If she gets lots of calls she may want to get a voice mail service.
She'll either need a little connector with a double phone line connector and a spare phone line cord, or she'll need to change the cord back and forth from Laptop to phone after each use.

But my parents only go online for 1-2 hrs at a time so they don't usually miss any important calls, so no biggie for them.