OK now, the Adobe Team replied me on Google Play Store :
Hi, Adobe Reader currently cannot save edits to files stored on SD Card for devices with KitKat 4.4. We are working on a fix for the same. A workaround is to save the file on your internal memory or on Acrobat.com to save edits. Thanks, Adobe Reader Team
I am glad that this is a known issue (i.e. not my own device went haywire) and that the developer is working on a fix already
Thanks