What cycles the player through the drop cycle?

Solution 1:

EDIT: Following the recent update as of 03/05/2016 the current chest cycle can be found HERE

Original Answer: The comments are right, and apparently also is the reddit post.

It is pre-determined when you get a drop by this updated list (to include new chests). I just got a giant, so I know exactly where in the list I am, and can start logging from there. Few things to note:

1. Having no free space, but wining a battle, moves you to the next chest. So In theory you can always get golden and above.

2. The list might get updated or the entire system scrapped by Supercell at any time, given they are the developers and can do whatever they please.
3. Having gone trough the list, it starts over from 9.

A win would push you forward in the cycle, regardless if you have a free slot or not. That's a win where trophies exchange place (eg. not against clan-member in a friendly match)

EDIT: What I've just red on this FAQ is having full chest slots and wining does not progress you on the list