Difference between “At the time” and “at the moment”

Question

What’s the difference between “At the/that time” and “At the/ that moment”? In which cases should they be used?

Answer 1

At the moment = now. "I'm in the garden with a cup of tea at the moment."

At that moment - "Mary unlocked the front door, and at that moment the telephone rang."

At the time = during the period when a past event happened. "I saw the Queen once. We were living in London at the time."

At that time is similar, but more likely to be used as part of a narrative. "At that time I had just moved to London."

Answer 2

At the time refers to something in the past. At the time there were no smartphones.

The second is present time. At the moment I am busy finishing my homework.