Some research led to this article from AllThingsD about how an Apple rep had confirmed that the service does indeed stream the music, but it also saves to the device, thus taking up storage space. It starts playing before it finishes downloading, but it does actually download and save.
MacRumors and a video from InsanelyGreatMac are somewhat unclear about this, but MacWorld agrees that on an iOS device, it's downloading (scroll down to "Playing iTunes Match songs on your Mac and on iOS").
UPDATE as per comment:
As of iOS 5, the device will delete cached data from apps when it's low on space. This caused some controversy at first because apps had no way to prevent cached data from being removed automatically. In the linked article, Marco Arment provides an example:
A common scenario: an Instapaper customer is stocking up an iPad for a
long flight. She syncs a bunch of movies and podcasts, downloads some
magazines, and buys a few new games, leaving very little free space.
Right before boarding, she remembers to download the newest issue of
The Economist. (I think highly of my customers.) This causes free
space to fall below the threshold that triggers the cleaner, which —
in the background, unbeknownst to her — deletes everything that was
saved in Instapaper. Later in the flight, with no internet
connectivity, she goes to launch Instapaper and finds it completely
empty.
This was fixed in iOS 5.0.1 (Apple Developer documentation) so that developers can differentiate between cached data that is and isn't important.
So, the short version: Once free space gets below a certain level, the device will regain whatever space it can by removing cached data from installed apps. If it can't do that, or if you're completely out of space, it'll probably just tell you that you don't have enough space to do what you're trying to do.
I think the difference is in the second paragraph that you posted. In your situation, if you have bought less then 7,000 songs from iTunes, then the rest counts against the storage.
In other words, it looks to me that only the songs that you bought from the iTunes store will not count against your 25,000 song limit. All other songs will (whether bought from somewhere such as Amazon, or ripped from a cd) likely count against that limit.
Keep in mind that you can have more than one Apple ID with purchased music in a library and according to the FAQ - only the Apple ID paying for iTunes Match/Apple Music is exempted from the 25k song limit.
It's not clear whether Family Sharing is all considered "one Apple ID" so you might want to be sure you re-download all family songs under your Apple ID if you are running up against a limit.
Best Answer
I have seen the same thing happen to my hard drive since enrolling in iTunes Match. Fortunately there are a couple of options.
One option is to select any songs you'd like to down-convert, right-click (or ctrl-click) the selection, and choose "Create xxx Version". This will re-encode the tracks based on the encoding options you have set within iTunes Preferences. Feel free to adjust these settings to the quality of your choice. Once they have finished converting, you may remove the higher-quality local copies. iTunes in the Cloud will retain the higher-rate 256kbps version, but your computer will now have a lower-quality version to playback from.
You can also remove local copies of songs entirely, by selecting them and pressing "delete". The songs will remain in your iTunes library as stream-able tracks through iTunes Match. Before doing this, though, I highly recommend backing up your entire library of songs to an external hard drive for safekeeping.
Hopefully between a combination of these two options you can reclaim much of your lost space.