If you went door to door with a petition trying to get enough signatures for a proposal to be voted on by citizens, what form of direct democracy would you be engaging in?

When a citizen goes door to door asking for signatures then they will be engaging into a form of democracy called: Initiative. There are two types of processes to submit an initiative: direct and indirect. The first one does not require signatures it goes directly into ballot, the second one requires an specific number of signatures according to the state's law in order to appear in ballot. Therefore, the democracy process described on the question is an Initiative with indirect process.

the answer is jerusalem