update: This solution had some weird side-effects, so I stopped using it. Leaving for historical purposes (and people who don't want to shell out a few bucks for the new solution I use.)
I figured it out: NoLine custom theme + UltraUXThemePatcher!
Upvote/comment if you'd like the files required with detailed instructions. (You can mess up your computer if you don't install the correct theme that matches your version of Windows.)
The system tray does only give more information (on the clock etc) because it has the space to do so. Please understand that the system tray is part of the taskbar and as such will resize to whatever size the taskbar is.
What you need to do is ensure that the taskbar is unlocked (right-click and make sure there is no checkbox in front of "lock the taskbar". If so, click it, to remove the checkbox.
Now, hover the mouse to the edge of the taskbar that is not touching the edge of the screen (the taskbar touches the edge of the screen on 3 sides, and the desktop on 1 side, you want to find this last edge.)
The mouse cursor will change to a resize icon. Now drag and hold the left mouse button and move the mouse as far to the edge of the screen as possible, making it smaller. It will move in intervals because the taskbar has fixed widths. Just move it as far as you can until the taskbar has the desired size.
For the best results, drag the taskbar in its center.
You may need to reorder certain taskbar elements. You can do this by dragging the | beam at the very left of each taskbar section (if applicable)
Once done, lock the task bar again by right-clicking the bar and select Lock the taskbar.
Best Answer
To get a 2-line date/time field:
Right-click on the taskbar and make sure "Lock the taskbar" is not checked.
Resize the taskbar to a single line of icons (smaller).
To get a 1-line date/time field:
Right-click on the taskbar and click "Takbar settings".
Set "Use small taskbar buttons" to "On".
When the taskbar is a double-row of icons it allows for 3 rows of data in the date/time field. Setting this to a single row of icons allows for a 2 row date/time field. Setting the icons to small sets the date/time field to a single row.