as stated
All electrical machines can generate and all electrical machines can motor (hence why I don't use the term motor or generator, just machine)
difference is in the flow of power. Some motors are actually really good generators AND that is a real PAIN when you are trying to control them to high-speed
When I use my hand to make the motor move, i.e. rotate the axis, the multimeter detects some voltage from the motor's supply pins. Can this voltage be used for any purpose? What is the key difference in its design that distinguish a motor from a generator?
Well thats not strictly true Every machine can be a generator and/or a motor (yes even a squirrel-cage induction machine). How they can generate however is different.
A squirrel-cage induction machine needs its windings exicited for it to start generating (ie AC applied to terminal but shaft span faster then sync speed == generation). Likewise a Switched-Reluctance machine can generate but it needs it coils excited in a certain pattern. Basically they need an external power to be able to then generate (as opose to a PMAC machine)
ALL machines can generate and ALL machines can motor. How they perform in the role oposite to what they were designed for is key to how you can use it in that role