Do people get those phantom pains with just a partial finger amputation?

Occasionaly: Sometimes, but often this is not a major problem and frequently resolves over time.

Nerve pain: Nerve pain form a nueroma may the better term. Phantom pain imlpies that the part is not there but the mechanismis similar. Sever injuries, crushing of tissue and nerve, stiffness, scarring can all lead to persistant pain. A nueroma means that there has been an injury to a nerve an healing is disorganized and swollen and scarred within the neuroma and they can be painful.