I have been lurking but finally registered. It's a holiday and then a weekend and most facilities aren't going to all the trouble of releasing people and shifting people until Monday, unless you're brought in by an ambulance with clear self harm or something concrete. You have a couple days to look for the bag. Search for it. If you can't find it simply forget it, take the approach you lost it.
Also you could text someone about going to a concert in a few months, say out loud about a trip you might want to go on in the summer. Ask people about what everyone is doing for christmas day, on your laptop look up these things that shows your planning or looking forward to the future.
Have a small diary and write different than you do that are positive things or mundane.
If it comes to them calling someone, say you've been arguing with family over moving, college or a job or dating things. If you seem fine and are 18 and they don't have anything except a piece of paper, you'll have a wellness check if that, you explain you had an argument or just seemed sad over the holidays b/c it reminds you of your grandparent who may have passed or sad over something small and they only have a piece of paper and nothing else they'll leave.
Police/Hospitals are short staffed on holidays with much bigger issues. If no harm is evident or seems evident or sure something evident nothing is happening. Just a worried mother who is overreacting because the past and you are fine and seem fine, a piece of paper is I highly enough, I don't see that happening since most places are full and if they have an open bed they probably are saving that for someone else who came from a hospital. Stay calm and rational act like the claims are made up and question her that she would go to so much trouble of faking a note so she can control you. You will be ok. Also if the paper or pen it was written on and unique. Put it in the garbage or a communal space.