I would never use "engaged" to mean "occupied".
If someone said, "The toilet is engaged," I would think "The toilet is getting married soon".
In the US, when we use "engaged" we almost always are talking about marriage.
The other meanings of "engaged" are really only used as technical language and not in typical conversations.
For example, in a factory, someone might say, "The mechanism is engaged" meaning "The mechanism is functioning".
In Korea, someone I was with pointed to a business and said, "I was engaged here."
I asked him in Korean, "Did you say that you were engaged to be married here?". 여기 약혼했다고?
He was surprised and he said no.
I told him not to use "engage" and told him to say "I used to work here".
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