5You are worrying too much about 'grammatically correct'. 'I died yesterday' is 'grammatically correct', but nonsense in most normal senses of the expression.
You are correct in your thinking:
"I have begun studying" (no time expression)
"I have been studying for 3 years"
"I began studying 3 years ago"
"I have begun studying since 2009" - This is possible only in the sense of 'I am in the beginning stages of a course of studies which began in 2009'. You could similarly say 'I have taken up yoga since we last saw each other.'