OK don't take this as a negative on you, because we all have to start learning somewhere, but you are doing it wrong. Here's the deal:
First of all, that single-speed chain appears to be an old-school type that does not have mushroomed pins, which means you can do the old-school method of pushing the pins out, and then back in again to re-section the chain. The more modern way is to use a master link, but I'll save that for later. For your chain, you went off the rails when you pushed that pin all the way out.
My chain does have mushroomed pins, except for that single black "master pin" that I figured was the one that I should be pressing out.
Yup, I pushed it right off the rails. lol
I did press it in just enough to clear the outer link and went to rotate that link out of the way and it wouldn't budge. I put pliers on it to try and rotate it and nothing. I thought it must be connected to the outer link on the other side, so I just kept pushing on the pin. I figured that I could push it to the other side until it clicked into the outer link, but there was still a lip in the way and I couldn't get the inner link out, so I just pushed the link onto the train tracks. lol
Once you do that, getting it back in again is possible but not guaranteed, and a nightmare to make happen.
Ohh, it was a challenge. lol
I must have dropped it 5 times when it came flying out of the chain breaker.
I ended up gluing a magnet to the table and stuck the pin on it while I figured out how I was going to get it back in.
In the meantime, I ordered some KMC single speed quick links, then thoroughly cleaned my chain with soap and water, brake cleaner and alcohol and gave it a hot wax treatment on the stove.
(I used my own plastic container because the kit was more money and came with half as much wax.)
I ended up using an adjustable pliers to press the pin back in, taking note of the direction, to install it the way it same way it came out.
I know a way to reliably do it but its not something I want to teach or you want to spend time learning. Instead what you do is NEVER let that pin go all the way out.
I'll keep that in mind, but I'm going to use quick links from now on.
Meanwhile, my chain is still holding with that master pin. I've put around 100 miles on the chain.
What you can do is practice on a sacrificial length of chain and do only a half-turn at a time. Count your half turns. As the pin gets further and further out, you start testing to see whether it is retracted enough to tease the chain apart. Once you find you can do it (there will still be a small nub of pin still sticking out on the inside) you remember how many turns it took to do that and file that knowledge away.
That makes sense. I didn't think of counting.
If you do have mushroomed ends on the chain, then what you do is twist hard and that lip will snap off as a natural part of the separation process.
That didn't look possible.
At least not with my piece of crap from Walmart. lol
This is the reason you do not re-use chain links after pulling the pins. You always re-section with master links. So if you are cutting a chain down to fewer links, thats fine. But if you are adding more back on you need to use a second master link, which is generally considered to be stronger than the actual original chain.
That's good to know.
I assumed the quick link would be the weakest link.
BUT I'm not sure that is true with the style of master link you have on that chain. A more typical style of master link is seen on this Z8.3:
What I see in your pic is sort of an old-school kind of master link and I wouldn't trust it the way I do the snap-lock kind.
That was a master link that I had in the garage for 25 years.
I thought I could possibly use it if I couldn't get the pin back in while I waited for my Missing Link to arrive.
And yes you probably will have an easier time in life with a better chain breaker. I use the Park Mini Brute which is very small and fits into a tool bag on your bike. Bigger ones are easier to use but the mini brute is all thats necessary to get the job done reliably.
I just ended up purchasing this,..
It will be here on Monday.
Thats a bad choice for a noob. Its great but is missing something: lacks the middle 'shelf' that lets you spread a link a tiny bit, and fix a stuck link which happens a lot.
The Park Mini Brute. Notice the two pins sticking up above the chain. Thats a second channel to put the chain so you can spread it a touch.
s*it !!
I think I'm missing a shelf. lol
EDIT: It looks like the Crank Brothers has a self,..
The crappy one has two shelves.
Maybe for a thicker motorcycle chain?
I made a hook out of a metal skewer from the dollar store.
Metal coat hangers are hard to find nowadays.