The second sentence does not look correct to me. The first and third are correct, but mean different things.
- I miss out on so many words when I'm writing something.
As you said, "miss out on" is used for opportunities. As MacMillan Dictionary says:
miss out on: to lose an opportunity to do or have something.
This would imply, perhaps, that you had an opportunity to use a good word, one that fit the context very well, but you missed it. Maybe you could not remember it.
- I miss so many words when I'm writing something.
This means that you forgot to write a word. It should have been there, but you accidentally did not put it in, or perhaps you didn't know that you should. I think this probably what you mean to say.
"Missing out" implies not doing something enjoyable or desirable, so it wouldn't be used for not doing something correct, like writing a word.
As for the second sentence, "out" without "on" is not used with "miss", unless it is intransitive: i.e. when there is no object to miss out on, one just "misses out." But "leave out" would be fine in this context.
I leave out so many words when I'm writing something.
Some people might infer that this implies intention, so you could also say "I accidentally leave out..." in order to clarify that it is unintentional.