What is a proper noun ('alam)?

The 'alam is definite by itself: say زَيْد and everyone knows which person you mean, with no الْ and no pointing needed. Names of people, places and tribes all qualify. Note one surprise: most proper names still carry tanwin (مُحَمَّدٌ، زَيْدٌ) — here tanwin is simply the noun's normal ending and says nothing about indefiniteness.

The word عَلَم ('alam) in everyday Arabic means a signpost, a banner, a landmark — something raised so it can be seen from far off.

عَلَمُ الطَّرِيقِ ('alamu al-ṭarīqi) — “the road marker”

As a grammatical term it means a name that identifies its owner outright, with no help from anything else.

If a name is already definite, why do some names carry ال?

Because that ال is not doing the work of definiteness. It has come along as part of the name and stuck to it. The word was definite before it arrived, so removing it would leave the name incomplete rather than indefinite.