MySQL COUNT with LIMIT
Solution 1:
This is actually how your query works and is a normal behaviour. Using LIMIT you will not limit the count or sum but only the returned rows. So your query will return n rows as stated in your LIMIT clause. And since your query actually returns only one row, applying a (non-zero) limit has no effect on the results.
However, your second query will work as expected and is an established way of solving this problem.