Thanks very much for bringing up these other examples, Yun.

I view this use of
own somewhat differently. In the sentences I offered,
own means this idea of what belongs to the subject and nobody else. It's used for clarity:
He watched his own money disappear in the fire. (not somebody else's as you might think)
They watched their own money disappear in the fire. (It's the same thing here; it was their money and not some other people's.
If we don't add
own in this example, we have an ambiguous sentence:
They watched their money disappear in the fire. Are we only speaking about one group of people or are we speaking about one group of people who saw some other group's money disappear in the fire?
In the sentences you've cited, I think
own is being used simply for emphasis, but not to bring clarity or avoid ambiguity. When you state your point of view in an argument or debate, you tend to emphasize as much as possible. I think that's the case in these sentences.
By the way, I hope in the first sentence you've cited that the subject was plural (
the readers) and that you have a typo. Otherwise, it's really ungrammatical in my estimation. The sentence should be either
"I imagine the
readers will not be offended if I repeat
their own terms in this place."
OR
"I imagine the
reader will not be offended if I repeat
his/her own terms in this place."