Difference between “at the time” and “at that time”

For example,

versus

Which is correct and why?

First, you identify the instance

That man robbed me. That is the man who robbed me.

Once the instance is identified, then the instance has been defined. Having been defined, means you have a definite way to point to the man. Now you can say,

The man first stopped me with a machete. Then he relieved me of all my jewelry.

Similarly, first identify the moment.

Kate: Joan, there was a time when we were kids and we a had a house next to a brook.

Joan: That time. I remember that time. At the time we were what ... ? Four, five years old ?

Kate: Yah, I was probably five, and you four. Do you remember the day our cat fell into the brook ?

Joan: That day. Yep, that day.

Kate: That was a pretty funny day, wasn't it?

Joan: The cat was jumping all over the brook trying to get out of the shallow water.

Kate: That week was also the week, when the utility company finally strung electricity to our house too.

Joan: Oh no, the days before that week, we had no electricity. We were like living in the wild.

Kate: We were living in the wild.

Joan: Actually, I really longed to live in the wild of Tennessee again.

Kate: I knew you had a wild streak in you.

Joan: What wild streak?

Kate: The wild streak that made mama pull her hair and gnash her teeth.

Joan: That wild streak?

Kate: Yes, that wild streak.

• I think you're saying that they both mean about the same thing, but it's not clear. – Barmar Jan 1 '16 at 3:29

When used as a non-demonstrative determiner, 'that' is synonymous with 'the'.

• Can you clarify by illustrating the demonstrative- and non-demonstrative-determiner usages of 'that'? – Edwin Ashworth Dec 30 '15 at 0:27
• I can't demonstrate the demonstrative usage via text... – AmI Dec 30 '15 at 1:22
• I don't think he/she understands the meaning of "demonstrative usage". – Blessed Geek Dec 30 '15 at 4:41
• I'm pretty sure it's a she. That/the beard is a dead giveaway. – Erik Kowal Jan 1 '16 at 11:22
• 'That thing' means the same thing as 'the thing', unless the speaker is pointing to something. {'That' is also a relative pronoun, so I had to specify its use.} – AmI Jan 7 '16 at 21:58