First of all, there's no reason that an agent has to do the *greedy action*;  Agents can *explore* or they can follow *options*.   This is not what separates on-policy from off-policy learning.

The reason that Q-learning is off-policy is that it updates its Q-values using the Q-value of the next state $s'$ and the *greedy action* $a'$.  In other words, it estimates the *return* (total discounted future reward) for state-action pairs assuming a greedy policy were followed despite the fact that it's not following a greedy policy.

The reason that SARSA is on-policy is that it updates its Q-values using the Q-value of the next state $s'$ and the *current policy's* action $a''$.  It estimates the return for state-action pairs assuming the current policy continues to be followed.

The distinction disappears if the current policy is a greedy policy.  However, such an agent would not be good since it never explores.

Have you looked at the book available for free online?  Sutton, Richard S., and Andrew G. Barto. Reinforcement learning: An introduction. Vol. 1. No. 1. Cambridge: MIT press, 1998.