Yes, 제가 is more formal and 내가 less formal (but in my opinion, still formal enough for Korean learners in most cases), but...
Have you studied enough to notice the subject markers 은/는 and 이/가?
And are you aware of 저 and 나? Both are ways to refer to yourself, 저 being the more formal one and 나 being the less formal one.
Well if you were to use 저 and 나 with 은/는, you'd have to use the ~는 and you'd get 저는 and 나는.
BUT, if you were to use 저 and 나 with 이/가, you'd have to use the ~가, but you wouldn't say 저가 or 나가, for whatever reason these would change to 제가 (저 + ㅣ + 가) and 내가 (나 + ㅣ + 가).
Hope that makes sense!
Rob...
p.s. They don't actually mean "I am", they just mean "I". It'll depend on what type of verb comes after to determine whether the "am" part will be there. For example, both of these subject pronouns could be part of the sentence, "I eat" or "I play".