Yes, you are correct. I am talking to you from amethyst, others are using plebstr, there are desktop clients too. You could also copy your secret key to another client and your user profile should load right up, just don't post that secret key to any of us.

We are also all connected to different relays too, I'm using the default relays for amethyst, damus has its own default relays. A lot of power users add extra relays, some are free and others are paid (for better speed etc.)

Is there a way to shorten the private key making it easy to remember? Else I would have to keep a copy of it somewhere. I am not a big fan of using any password manager.

Or does a short private key beats the purpose as it would be easier to social hack it?

Mmmm, its not something that could be practically done by a user. There have been some discussions about changing the format, but that's happening at the client or protocol level. Someone smarter than me could probably make a "seed phrase" out of it that's easier to remember or write down, like how bitcoin seed phrases work, but it wouldn't be shorter. The length of the key is important so that it can't be physically hacked by a strong computer.