As to what makes a glee a glee, I dunno... but as I was taught, a catch is a round whose words make one sense in single line, but when all parts are put together make another sense altogether -- usually bawdy. The term "catch" may refer to the broken rhythm created by the oddly-place rests in each part where the interplay happens.

I'm not familiar with your example, but one by Peter Shickele (as PDQ Bach) will illustrate:

I
Please, kind sir, that portrait I see
If that's your daughter present her to me;
Look! (rest) her (rest) face could launch a thousand ships (2x)
Thousand ships, thousand ships...
II
Very well, it can be arranged if you will please
Sit you down, make yourself at home while she's
(rest) Up (rest) dress - ing, she'll be down in a jiffy (2x)
Jiffy, jiffy, jiffy...

Innocent enough, but when the parts are combined, the third lines form
"LOOK up HER dress" which never fails to elicit howls from the audience. Juvenile, yes, but evidently a time-honored musical form!

Use this info at your own risk as it is "undocumented" and may be a figment of my often unreliable brain.

-HM