I worked at a public agency and would have different out-of-office messages for internal and external. I was chastised for having a “too informal” message- because the idiot talking to me didn’t realize me saying “I’ll be back next Tuesday for the big staff meeting” (or whatever) was just for co-workers and not the public. I told them but of course it didn’t matter. So from then on I always made sure to start my internal OOOs- “Hello Company X comrade…. blah.” So it was clear which was which. I am not able to respond to your email promptly because my husband died. I will not be accepting zoom invitations. Please do not respond by suggesting future alternative dates. I don’t know when I will be able to speak without crying.
PS: We love our downtime and here at HAR, we're big believers everyone needs a vacay from work. Yes, even you—the social media junkies, the workaholics, the entrepreneurial diehards who accidentally (or "accidentally") take their smartphone into the shower.

A. All faculty and staff are encouraged to leave their work areas clean, including taking food home. Also, please close all windows and doors, and shut down computers, monitors, printers and other similar equipment (except LAN servers and network devices). Portable space heaters, coffee pots, fans, radios and other non-essential equipment should be turned off and unplugged. If you notice any water fixtures that are leaking or dripping, or any other maintenance issues, please contact Facilities at [email protected] as far in advance of winter break as possible so these issues may be addressed appropriately.
I think that’s maybe something you need to deal with internally with the person/people who you’re asking people to speak to instead – I think if you start to ask the original sender to update you or cc you then it’s going to start to annoy people that they are doing the running around, plus not everyone will do it.