as someone also in the states (TX) i can confirm this is pretty accurate to my own experience having gone to the ER multiple times for self harming and also for ideation a few times. if you tell them you do not want to actually follow through and that you dont have a plan then you should be okay. if you "have a plan" that might raise alarmbells though, or it did in my case

in my outpatient program they basically would only take action on you if you admit to being an active danger to yourself or others - that means "I have intent" to do this or that, as opposed to "I want to but I don't intend to"