If you are purchasing a bicycle with the expectation that others will use it and you will in turn use theirs, then I'm not sure the term "personal bike" applies.

It's more like your personal contribution or donation to a collective scheme. This seems like what this article describes.

That's basically what some student have been doing here for decades now, though not out of idealism. The disadvantage is that sometimes a local entrepreneur comes by to sell you your own bike back for 20 euros cash.

Blindly fumbling through the night, you bump into something foreign. Feeling it, brushing your palms and fingertips across it, you realize what it is: the fundamental contradiction of private property.