I know this because I played coin rooms when i first started a few years back. I haven’t played them in quite some time, but the format remains the same;
Each of the divisions you listed has 3 levels, conveniently called I, II, III. Can’t remember all of the division names, but you have the majority of them except for the Diamond Division. There is a progression and I believe you have it correctly listed with Diamond being the top Division. Each Division level has a payout in coin prize money for the top ten players of the previous week. The higher up you move the greater the coin prize.
If you hit the PLAY NOW button you will bring up the Coin Room Selection page. At the very top of the page is a segmented line. Hit that line and your coin room status, if you play, will be displayed to include your division, where you rank in that Division/Level, and the amount of coin earnings you have to date for that week. You will also see the screen names of all other players in your Division/Level and their ranking based upon their coin earnings. On Monday you are assigned a Division/Level based upon your prior weeks performance and you start the week from that position and begin to accrue coin earnings again.