Judging by the timing of it happening, i'd imagine it is the battery in both these instances. My Mk2 Punto JTD started doing this at 6 years old when the weather turned cold and the battery was wearing out. The column did fail in the end, but not until 8.5 years old - the week after i sold it, oops!
The Stilo (and new Bravo) has a different make of steering column and failure of these is virtually unheard of. It's the Delphi columns on Punto's & Panda's (which are also used on 500s) that are known to cause trouble, though it's Mk2 Punto's and Grande Punto's that seem to fail most. Punto Evo's and 500s could follow suit, they're still abit new to tell, but the Panda has been out long enough and that seems to be better than the Punto's, so the columns may be improved.