GHNeko: that can be answered by looking at how these three terms interact with each other (particularly the first and third).
"Wii Network Service" means the service whereby we can distribute Content, messages (including commercial and non-commercial messages) and other data to your Wii Console, or link you to an Internet site, and whereby you can send and receive Content, messages (including commercial and non-commercial messages), and other data, including Third-Party Data, using your Wii Console. The "Wii Network Service" includes, but is not limited to, the Wii Shop.
"Content" means the software, services, material and other data which we own or have the right to use, distribute or sublicense, and which we send to, or make available as a download through, the Wii Console or the Wii Network Service.
"Third-Party Data" means Content or other software, services, material, messages (including commercial and non-commercial messages), links to Internet sites, and other data either created by, or sent to, your Wii Console through the Wii Network Service by any non-Nintendo entity or person (including you).
How loosely do you want to interpret this? An
extremely loose interpretation might allow it.
Maybe. "Third-Party Data" actually uses the "Wii Network Service" as part of its definition, and I'm pretty sure that the data used online is not defined as "Content," being simply button presses and
user-created content (thus not fitting the strict definition of Content provided in the document), so this rests most of the burden on the definition of the "Wii Network Service."
Unfortunately, you
are sending and receiving "other data" using your Wii Console. It's a catch-all phrase, and by the other terms surrounding it, one can infer that it includes online play of games. The Wii Network Service appears to encompass all data either sent or received by your Wii through the internet.
Also, if you wanted to use WiiConnect24 as a measure of if something counted as being Wii Network Service or not... It mentions internet sites in there, and the internet channel is accessible even when WiiConnect24 is off. So just because online doesn't require WiiConnect24, that doesn't necessarily mean it's not a part of the Wii Network Service.