I don't think he's very mature; he's still unnecessarily frightened by many things.
I don't think fear is always a factor at how mature someone is. Well if it is unnecessary fear then it might count as immaturity. But I haven't seen Petrie being "unnecessarily afraid" that often. Probably the only time I recall seeing him like that is during that episode of Land Before Time, when he was afraid of "bad luck".
Things Petrie's been afraid of almost always seem to come with a good reason. For example, the refusal of flying over to get help in LBT5, this is understandable. They had never seen the ocean before. It was uncharted and unexplored territory. They had no idea what lived in those waters, and they didn't know if they were safe even in the air. This is probably why Petrie didn't fly over.
Being mature means being responsible, and making the right choices. It also means having good judgement and knowing when to back down, and knowing when things are too dangerous. I believe Petrie can be mature when he wants to be, but can be immature at times since he is still a kid.
I sometimes see Petrie as being, well, more mature than the others at times, such as in The Canyon of Shiny Stones, referring to the part where he refused to go because of Smokey Mountain. He had the better judgement of the group, and made the right decision in trying to stop the others from going. This cautious behavior made him, in my opinion, the most mature of the group.
If Petrie was more mature, then he'd stop saying "me this" or "me that".
I've never considered bad grammar as a sign of immaturity. Petrie does say "I" sometimes, but it is a rarity. One of the few times we hear him say "I" is in the original Land Before Time, when he says "I-I-I fly?" and "I fall?!" So Petrie does know the "correct" way to talk; he just chooses to talk his own way. It's like in Land Before Time 12, he just likes to do things "his way".
