How to get iLife '13 for free (update from iLife '11)?
You thought right, the update to new iLife is free.

However, Apple had dropped the "iLife" name a while ago, now the apps are just being called iPhoto, iMovie and GarageBand.

To get the new versions of the apps, follow these steps:


Upgrade to Mavericks via App Store. This requires at least OS X 10.6.8 to be installed.
Change system preferred language to English and reboot your Mac.
Open App Store and search for GarageBand. It is free now, no matter if you had iLife before.
Go to Updates tab and wait for a minute. Newest versions of iPhoto (9.5) and iMovie (10.0) should appear.


Note, that even pirated and demo versions of Pages, Numbers, Keynote, iMovie, iPhoto and Aperture can be upgraded via App Store. They will appear in Purchases tab a few minutes after the upgrade.

Troubleshooting


If you do not see a response to Update button click, check your
Launchpad. Upgrades are downloading in background.
If you are getting an error about wrong Apple ID when trying to
upgrade, try installing OS X Mavericks in a virtual machine
(VMWare Workstation or VirtualBox) and repeating the fourth step.