I'm not sure if I fully understand the question.
The only "normal" way to increase your population is by clicking that button and spending the appropriate amount of food.
Every other way requires a card action. Some examples are:
Frugality - still costs the same amount of food, but you click on the card (and then receive bonus food back).
Inhabited Territory - when this event comes up, the winner of the territory gets some free population.
Ocean Liner Service - a wonder that let's you click on it once a turn to increase your population once you've finished building it.
Barbosa - a leader that let's you build a military unit directly (increases population and build the unit all in one move) by clicking on him.
Then there are other various events that can get seeded at political actions that can increase (and/or decrease) population.
I may be forgetting some other instances, but those are the most common ones I remember.