They are pretty standard really, and maybe in your case, it is being used to grease the wheels to get you admitted. As it is formally down you are a risk to yourself and apparently others. That makes it harder to fob you off and ignore you so the bean counters can't quite so readily deny you a bed you need. Often frontline services are in conflict with what management want. In these risk assessments they also look for beyond what you are just saying. They sometimes use wording and body language as a means to take a risk on you still being alive so they can get a bed for someone who wants to gouge out their sons eyes. Because if they don't, demons will eat their sons soul, which is worse than their son being blind. It is all about triage these days. The cynic in me also thinks its about what will make a bigger splash in the papers and invite greater backlash. Bed shortages is also why getting voluntarily admitted is so hard.
Did they ask you about how you would keep yourself safe? Did they also get you to sign a contract of safety?
It is a bit concerning that you signed something you do not agree with. You don't want your notes undermined by false information. They may well have examined notes taken on you from the past. Or taken statements from others and used what they have said. But if you dispute that assertion make that clear and ask how they came to that conclusion? Then put them straight.
Regardless of what is happening it may well be worth trying to get an advocate. Bureaucratic systems drowning in top-down pressures can sadly lose sight of the needs of the client. As you rightly recognise they are underfunded and being meddled with by politicians to the point of collapse. It is the worst I have ever seen it... Which is of no help to you or anyone else for that matter.
You could look for your local Mind.
https://www.mind.org.uk/information-support/local-minds/
Ring them and see if they have advocacy available in your area or know how you can access some from elsewhere. They can be pretty good at applying pressure on your behalf.
Lastly it may be worthwhile grabbing yourself a digital voice recorder. Then when interacting with any professional ask if you can record them because your memory is crap and you don't want to forget anything important. Most should be fine with that. But if you encounter someone who says no. Simply state, "I thought I could under disability rights?" Then enjoy watching their facial expression change. The reason I suggest this is because assholes don't like scrutiny and if they feel scrutinised they are less likely to cut corners at your expense. It also means you have a record of what has been said or promised. Evidence in this era is king when it comes to getting what you want and making disinterested people behave.
Really wish you all the best amidst these broken systems.
Peace.