Only one from those proposed answers could be even close to something resembling HRP.
In short HRP on SS13 servers is not acting (in character) like anonymous all-knowing online troll with ADHD. Pretty much it. How far you are on that scale is defined by how little you act like one.
You have LRP, where you just do whatever you want, mainly about only you having fun by “beating” other players in one way, or another, most of the time by somehow preventing them from playing (clicking a dude with toolbox till horizontal being the most common example, but stunning, cuffing, stripping naked and then shoving into a small room with bolted/welder doors counts).
On the opposite, the object of your search, is HRP, where you often put other people’s enjoyment above your own and gain “fun” and enjoyment from that.
Lets say you play vanilla SS13 and you notice a cultist going out of the maintenance for a while. LRP player would click the cultist till horizontal and/or call the sec by just saying “cult in “X” maintenance” on comms. MRP player would call out a weird guy in a hooded robe on comms. HRP player would start a conversation with cultist about stuff, or pretending to be scared a bit, depending on the type of character he is playing. There is also fourth type of RP, which I call “Zero Role Play”, in this situation ZRP player would not do anything, but continue with whatever he was doing at the time. Silent PFC who just shoots xenos and says nothing but “*medic”, or calling out Queen, or Xeno who just fights and says pretty much the same as PFC, those are ZRP players and they are perfectly fine.
“Quirky” gimmicks, typing quirks, making people laugh, using a lot of emotes and being expressive, all of that are tools that can be used both in LRP and HRP way. On CM most often than not, they are used in LRP way.
“Vanilla” HRP servers tend to put a lot of pressure on “realism” and “realisticly”, life on a space station would be boring. The way SS13 allows players to communicate doesn’t help either, everything goes trough chat box. GMod RP servers have it better because you can act in more ways that writing text and then holding ctrl while pressing arrows in order to spin like that toy for autists that was popular few years back.
I personaly think that it is impossible to achieve fun HRP with random people (who have no stakes in the game) without heavy moderation.
Those “boy’s only club” servers, those that shall not be named, get away with it, because to hop on them, you had to put some effort and gain trust. CM has certain roles whitelisted because whoever made them that way in the first place understood it.
PvE is a fine example. You gotta be luckey to roll and get a chance to play and if you act stupid, GM just makes you disappear, so everyone atleast tries to not act like a spoiled teenager with ADHD chock-full of hormones. There is also no pressure from the players on the other side who are not hold to any RP standards other than not using net-speak in chat.