What would characterize "zones of influence" is that, legally, the territories dividied would not be ruled by the colonizing power. They would leave a nominally independent local power, but that power would be subordinated completely to the local representative (embassador, etc.) of the foreign power.
It was a form of "soft" colonization, which would ensure the colonizing power that natural resources and trade of the region would be in the hands of its companies and individuals, while not meddling with the internal politics (unless the local administration wanted to assert its independence, or was in danger of being topped by an hostile -to the foreign power- faction).
The recognition in a treaty of the zones of influence was meant to be a way to prevent the powers signing the treaty from trying to extend their own zones of influence at the expense of zones recognized to other power.
For example, with the treaty of Sevres, SW Turkey would have been legally a part of Turkey, but mining operations, critical infrastructures (like ports) and trade would have been controlled by the Italian government (which would have handed them to Italian corporations), and France could not make any intervention in that zone. That ruled out the possibility of the Turkish government playing them one against the other.