An ordinary person would just say "Jem". In Polish we have something called "implied subject", when we drop the subject (only when we'd normally use pronouns, I believe), which means that while we know it is there, we do not say/write it, unless we want to emphasise that we are actually doing it. So you would use it in such a situation: A parent and two children sit at the table. The parent says "eat, slowpokes!" and one of the children exclaims "Ja jem!" (I am eating), looking suggestively at the other child. On the other hand, imagine a person calling their friend, asking "what are you doing?". The friend would just answer "Jem", because emphasis would not be needed.
From my opinion, "Ja jem" and "Jem" have the same meaning. But sounds like they have different emotions.
"Jem" is very clear for explaining who is doing. So, "Jem" is like "I am eating"; "Ja jem" is like "I + I am eating". Sounds like the speaker wants to emphasize himself/herself. Of course, if the speaker is saying "Ja jem" by very gentle way of speaking, I would not feel he/she wants to emphasize himself/herself so much.
In polish there isn't present simple and present continues time. There is one time present. This polish time we use when we want to speak of the action "now" and action repeat regulary. Well, in one polish presen time are two english time. So, "Ja jem" cen be speak "I eat" and "I eating" but "I eating" more fit to "Ja teraz jem" (teraz = now). I hope that you understand ;)
Ps I just eat = Ja właśnie jem. This is the same as "I'm eating at the moment"
An English-speaking Pole will of course understand what you wrote, but none of those is a Polish word. We pronounce Y in a completely different way. This seems to be accepted 'with a typo' (or rather with two), which is surprising... but I guess we can't do anything about it.