Since Snow Leopard upgrade, Time Machine broken on both laptops

Question

Concentrating of the "first" laptop, the Dashboard log widget shows:
Starting standard backup
Attempting to mount network destination using URL: afp://***@10.0.1.1:548/Time%20Machine%20A
NetAuthConnectToServerSync failed with error: 80 for url: afp://***@10.0.1.1:548/Time%20Machine%20A
Backup failed with error: 19

Answer 1

In addition to Kappy's list, try turning TM off, then renaming your Computer (System Preferences > Sharing). Keep it short, use only letters and numbers (omit spaces and punctuation). Then click the Edit button and make the corresponding change to your +Local Hostname.+
Make similar changes to your Airport and Network names via Airport Utility (and I'd not recommend calling your AEBS a Time Capsule, just to reduce future confusion by others reading your logs).
Connect the external drive directly to your Mac. When it appears in the Finder sidebar, right-click it and change it's name similarly as well.
This makes no sense, but often works anyway.
It doesn't look like your backups are corrupted, or it's not getting far enough to send that sort of message, but it's possible. So, while the disk is connected directly, run a +*Repair Disk+* (not permissions) on it anyway.
Then turn TM back on and select the (new) TM name. You may have to open it via the FInder to get it to appear in the selection list.
It must be noted that doing backups this way is not officially supported:
http://support.apple.com/kb/HT1733
http://support.apple.com/kb/HT2038
but many of us (including me) are doing it anyway.