The game might be unofficial, which we don't cover in this series.
First, have a look at the Tetris Fangames series and see if your game isn't represented there: https://www.speedrun.com/fantetris
If it is an official game that you're missing, leave a comment and we'll have a look.
The primary mode of time-based competition in the Tetris scene is undoubtedly the 40 lines mode, or Sprint as it's called in "official" terms. For some time, NullpoMino, a certain Tetris-clone, has also (sort of) been the de-facto game in which people compete.
Having been a part of the community for a while, and as a mod for NullpoMino here on speedrun.com, I've been noticing that there are other implementations of the game that people enjoy playing, specifically clones such as jstris, and Cultris II as well as Tetris Online Poland (private server of the now defunct Tetris Online Japan, and not a clone). To some extent, Tetris Friends and Puyo Puyo Tetris.
These games, too, have a 40 line mode, and people are achieving times somewhat equal to what they can achieve in NullpoMino. The current leaderboard of 40 lines mode on HardDrop does not discriminate between any of these versions, as long as the community deems it acceptable. Sure, some versions are imposing slowdowns compared to others by introducing animations and delays on line clears and piece placements, so there is no direct 1:1 comparison between top times in a game such as jstris and Puyo Puyo Tetris, but at that point it's up to the user to determine whether they want to compare themselves to the greater Tetris community, or just in their respective game.
What I'm trying to convey here is that the the NullpoMino leaderboard feels too limiting as other games are equally relevant in the competition. Would a general Tetris 40 line leaderboard be at all possible, seeing as it's not a specific "game" anymore?
I don't know how useful this would be for anyone, but I found Tarzan to be much more nice looking and easy to play by using DxWnd to force window-mode and using "Software High Res" as the graphic setting.
What I mean, is that you could go from the pains of this
To something more like this
The ingredients required to conjure this witchcraft is DxWnd: http://sourceforge.net/projects/dxwnd/files/Latest%20build/v2_03_32_build.rar/download
Not surprising, it's a tool that hooks into the executable and forces window mode. As far as setup goes, you extract the .rar file into some folder and run dxwnd.exe. Right click in the white area and select Add, or go to Edit->Add. I'm not sure if I went with any distinct changes from the default settings, but for posterity, here are my settings. I'm fairly sure I haven't changed anything on the other tabs.
After that, press OK and doubleclick your entry of Tarzan to start the game. It'll launch in fullscreen, so Alt-Tab to put it out into its usual "Software Window" mode. There's a good chance it's gonna end up looking like this.
In the case it does, let it get to the main menu, press F2 and go to the Graphic settings. On the off chance you're unable to read anything, navigate with the following sequence from the main menu.
F2, arrow down, arrow down, arrow left, enter. It'll show the DxWnd splash and get you to the magical lands of "Software High Res".
To capture this in OBS, select Capture Window and make sure to check Entire Window. Tarzan will spaz out a bit when you tab out of focus of the game, but it's not too big of a deal to press F2 and select the right graphic setting again. This is how I've got it set up at the moment, I reckon it looks alright!