I'm not a teacher, but I would say that "on purpose" is the most casual of the three. It's the phrase one would expect to hear from a child complaining of some injustice - "You did that on purpose".
In everyday informal settings, I'd be most likely to hear and use "deliberately". My wife will often say "that car deliberately sped up as soon as I had a chance to pass, then slowed down again". In a sentence like that one, "intentionally" would sound a tiny bit unusual, to my ears. I use "intentionally" most when writing. For example, someone once thought I had written "contemn" instead of "condemn". I replied that I had intentionally used "contemn", because it was my intent, it was the word I had intended to use.
Very interesting. I thought that "on purpose" was the most formal of all the three(*). So, if I choose to say "on purpose" I will be OK, right?
(*) Please, is this expression correct?
I would have said "the most formal of the three." or, "of all of them, it is the most formal" or "of the three of them, it is the most formal".
You would not be incorrect to say "on purpose". I do hear and read it in adult conversation, it's just that in NZ English, "deliberately" would be used more often, I think. That's my assessment, as someone who is not a professional teacher.
does these expressions mean the same, please?Yes, they do except for one example I want to give.
"Calmly and deliberately (= slowly and carefully), she poured petrol over the car and set it alight". "Intentionally' would not be suitable in this context.