$2^m$ requirement is for the pre-image attack. You have to hash approximately $2^m$ messages to find the message that has the same value you were looking for if the hash function has pre-image resistance.

In a collision attack, you are looking for two messages that have the same hash value.

If you look at Wikipedia Birthday Paradox at section Cast as a collision problem, it is also generalized for the arbitrary number of days not just for 365 days. The approximation is given by

$$n \approx \sqrt{2^m}$$

$2^m$ requirement is for the pre-image attack. You have to hash approximately $2^m$ messages to find the message that has the same value you were looking for if the hash function has pre-image resistance.

In a collision attack, you are looking for two messages that have the same hash value.

If you look at Wikipedia Birthday Paradox at section Cast as a collision problem, it is also generalized for the arbitrary number of days not just for 365 days. The approximation is given by

$$n \approx \sqrt{2^m}$$ and we call it generic birthday attack.

Note that: if you find an attack which is faster than generic birthday attack like SHA-1 attack you will be famous.