Is the term that represents citizens formally and officially requesting their lawmakers to make a certain law.

I believe the answer is: initiative

In political field, initiative usually started by increasing an awareness toward a certain cause that the voters want. This usually be done by things such as collecting large number of petition or making famous media campaign. The representatives then would see this and see it as an opportunity to obtain support, so they abide to the initiatives in order to gain favour for the next election.

in an absolute  monarchy, the  monarchrules as an autocrat, with absolutepower  over the state and government — for example, the right to rule by decree, promulgate laws, and impose punishments. the  monarch isnominally sovereign but the electorate, through their legislature, exercise political sovereignty.