I chose to update from 5.1.1 to 6.1 from the Settings app. Right now, the iPad only displays the picture that says I should connect it to iTunes. After I do this, I get a dialog box "iTunes has detected iPad in recovery mode. You must restore this iPad before it can be used with iTunes". I understand this procedure erases all data and customizations, and that's clearly not what I'd want to do (it's like having to destroy the house if you just want to paint a few rooms). Is this a part of a an iOS upgrade nowadays, or is there any way around it?

You cannot update a jailbroken device using the Over The Air (OTA) mechanisms. It will fail.

I trust you backed up the device recently, ideally before you started the process?

If so, then you can let iTunes restore the device and then restore your backup data, apps, etc.

Yes, unfortunately I tried OTA first. I used iTunes later on, but after downloading iOS update, it claims "The iPad could not be restored. An unknown error occurred (3014)". Nothing of help on their website if I click More Information button.