This is how I finally got Time Machine to work over WiFi:
- I plugged my big-ass[1] third-party hard drive into the USB port of the Airport Extreme base station that I had purchased the day before.
- I let my desktop Mac, connected to the house LAN via Ethernet, discover the base station automatically (it shows up under the "Shared" section in any Finder window's left sidebar). Via the Finder, I connected to the airport and mounted the drive.
- I told the desktop Mac's Time Machine System Preferences to use this mounted drive as its disk, and let 'er rip. Hours later, I had a working, browsable Time Machine history on that machine. Hooray!
- I set up my laptop beside the base station, connected to it directly with an Ethernet cable, and turned off the laptop's Airport access in order to ensure that it would use only Ethernet for the time being.
- I repeated steps two and three with the laptop, mounting the backup drive via the network and letting the laptop spend a few hours making its initial backup. And then it worked too. Hooray!
- I turned the laptop's Airport back on and unplugged it from Ethernet, so it's back to how it usually is. Time Machine continues to work as nicely as you please, both in making its hourly incremental backups, and in browsing them through the Time Machine application.
Previously, I had the hard drive plugged directly into my desktop Mac's USB, and tried to have the laptop back up to it via network-mounting it from there, but it didn't work properly - Time Machine would make all of its scheduled backups, but the Time Machine application would not recognize them, acting as if no backups had ever been made.
Admittedly, I changed several variables at once going from there to my current, working setup. Most obviously, there is the presence of the new Airport base station, and the fact that the backup drive is now plugged into it rather the desktop Mac. But also, I didn't know that my laptop's ethernet port worked - I thought it was broken! So I didn't try to make my initial backup that way; instead, I connected the hard drive to the laptop via USB, performed the backup, and then reconnected the hard drive to the desktop Mac, doing subsequent backups via the network. I suspect this may have confused matters. (I discovered the working state of the laptop's ethernet last night, as a desperate move. Wow.)
Therefore, I can't prove that my previous setup wouldn't have worked if I had tried to connect my laptop to the network differently. However, I don't regret purchasing the Airport, because the house now has a very nice new router. It gives us not just 802.11n WiFi (several times faster than anything we've had before) but also gigabit ethernet (ibid) and a separate wireless internet node for guests, with a trivial password and no LAN access. Plus,
classicaljunkie's old router is junky and prone to freeze up if you looked at it funny, so just as well that it get replaced.
[1] Yes, I'm aware that calling one and a half measly terabytes "big-ass" will seem laughable in a dozen years' time. I well recall how I purchased a two-gigabyte hard drive in 1997 and how infinitely huge it seemed then, and on and on back through time. This setup is for today, and lo, the ass, it is big.