You're correct.

As a native English speaker I was aware of the older meaning, however as someone that often "quickly explains english" to ESL folk and|or non-Commonwealth English backgrounds I went with "tearing paper" as a starting point as it felt more likely to be familiar.

This is consistent with your last paragraph re: modern usage.

I've got a much thumbed multi volume OED edition on my shelves .. it's boggling how many words have half page or more entries with multiple meanings and historical backstory.