People do not normally rent out their own private vehicles to other people, so I would not understand that to mean that someone accepted money for someone else to use their private car. If it were the case, I would expect something like "I rented my car out for $100."I rented a car for $100.
A car lease is for a long term and contains other sorts of provisions. Cars can be leased for business purposes or a private individual can "lease" a car rather than buying one.
You would rent a car for a week while you are on vacation, or in another city for business.
You would lease a car for 2 or 3 years to have steady payments, minimal maintenance costs and to always have a newer vehicle.
Apartments (flats) are technically leased, but most people speak about "renting" an apartment.
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