I'd say it's a non-defining relative clause: it adds extra information which isn't necessary for the clause to be grammatically correct. In fact you could go on with something like this:
"… she was involved in a relationship with a terrific man, "WillW, which lasted about three months, but which ultimately ended when Will got married."

The sentence:
"Jane has had numerous affairs over the past several years after her third divorce, and was involved in an "intimate relationship with a terrific man, "Will", that lasted about 3 months."

Is "that lasted …" a defining clause in this sentence?
"That" is used, so probably yes, but I'm not sure.

In simple examples it's easy to understand which type the clause belongs to.

It changes the meaning, because it adds extra info:

The people who saw the accident were shocked. (Only the people who saw the accident were shocked. The other people – who didn't see it – weren't shocked.
The people, who saw the accident, were shocked. (All the people who saw the accident were shocked.)

Because the way I am thinking; that when we remove the relative clause incase of "non-defining" the meaning is not changed , rather than in defining clause removing the clause change the meaning.

Please would you correct me if I am thinking wrong? or what does the last pargraph- in the lesson- mean?

Thanks alot Clare! you are the herio :-)