The distinction is the entire message, and it is very hard to make without sounding like you are angling for an invitation you have not been offered.
There is a specific state where you do not want to be alone and also do not want an evening. You are not lonely exactly. You would just rather somebody else were in the room, or on the phone, or reachable.
Adult social life has almost no vocabulary for this. Everything on offer is a plan: dinner, drinks, a call scheduled for Thursday. All of it has a start time, a duration and an implied agenda, and all of it costs more than the thing you actually wanted.
A plan is an event. It is anticipated, prepared for, and performed. You will be asked how you are and you will have to answer properly. It has a shape and the shape has to be filled.
Company is a state. Nobody performs. There may be long gaps. Something else is usually going on — cooking, driving, a game on in the background — and the talking is incidental to being in the same space.
The second one is what people mean by "hanging out", and it is the thing that quietly disappears from your life somewhere in your twenties. Not because anyone decided against it, but because it has no calendar entry and everything competing with it does.
Because every phrasing sounds like it wants more than it does.
"Are you free tonight" proposes an evening. "Do you want to do something" proposes an activity. "Can I call you" implies there is a subject. Ask for company and the other person reasonably hears a request for a plan, and either produces one or apologises for not having room for one.
There is also a status problem in it. Asking for a plan reads as sociable. Asking for company reads, at least to the person asking, as needing something — and needing something is the part most people will do a lot to avoid saying out loud.
The trick in all of these is to name the low stakes explicitly, because the default reading is always the higher-stakes one.
Attach it to something you were doing anyway.
making dinner, want to be on the phone while I do it
I have got an hour of boring admin, fancy being on a call doing your own thing
Say the quiet part.
not suggesting we do anything, just want company
I do not need to talk about anything, I just do not want to be on my own this evening
Offer presence rather than asking for it, which is the same thing and much easier to send.
I am around all evening doing nothing, call me if you want
Make the exit obvious. Half of why people hesitate to accept is that they cannot see how it ends.
ten minutes or two hours, whichever
The version that works best is usually two people doing separate things in the same space or on the same line. A call left open while you both work. Somebody reading on the other end of the sofa. A drive.
It works because it removes the obligation to generate conversation, which is the actual cost of most socialising. The talking happens in bursts when there is something to say and stops when there is not, and neither person has to manage the silence.
It is also the format most people last did properly when they lived with friends, which is part of why its absence feels like something specific has gone rather than a general drift.
Sometimes there is nobody available for it, which is an ordinary scheduling fact rather than a verdict, and there is a wider list of what is open when that happens.
Worth saying plainly where we sit in that: Teylo is four people you text or call, and one of them will pick up and talk about nothing in particular for as long as you want. That is genuinely the shape of company, and it is genuinely not a person on the other end of the sofa. Both of those are true and the second one does not stop being true because the first is useful.
If asking for company feels harder than asking for a plan, that is usually not shyness. A plan can be justified — there is a reason, an occasion, a thing being done. Company cannot. You are asking somebody to spend time on you with no return offered.
Which is, when you look at it directly, what friendship consists of almost entirely. The plan was always the excuse.
Wanting somebody nearby with nothing required of them is not a lesser version of wanting to see people. For most of human history it was the ordinary form, and the version that needs a date, a venue and a start time is the recent invention.