I just fixed it. To do so, I had to remove all of the installed software using npm and manually remove the node_modules directory. After a clean re-installation from a pristine state, it started working (almost...).
It looks like you have to remove the node_modules directory by hand because npm does not remove it and it does not correctly install all of the required software from scratch if it find an existing node_modules directory on your hard disk.
Now, I'm dealing with a path-related error raised by Apache Cordova:
$ sudo ionic run android
ERROR: Error: Failed to find 'ANDROID_HOME' environment variable. Try setting setting it manually.
Failed to find 'android' command in your 'PATH'. Try update your 'PATH' to include path to valid SDK directory.
Please note that:
$ sudo echo $ANDROID_HOME
/home/alex/Android/Sdk
$ sudo which android
$ (empty string)
$ which android
/home/alex/Android/Sdk/tools/android
As long as I can see, this new error is due to the fact that I have Android Studio installed as a regular user (alex) and Apache Cordova/Ionic Framework installed as super-user with sudo.