In "It has rained a lot since June," when did the rain occur?
And in "It has only rained twice since June," when did the two rain sessions occur?
Could the said rain have occurred in June?
All that you can tell is that it has rained a lot after June ended in the first case and that it has rained only twice after June ended in the second case.
The sentences have an implicit meaning of "since June ended." That implicit meaning is idiomatic. It is true that when we allow implicit meanings that there is no explicit clue that precludes "since June began." It is simply an idiom.