Leased vehicles typically come from car dealerships, whereas rented vehicles normally come from car rental agencies in standalone stores or at airports.
A lease is an agreement to buy the vehicle and the period is at least one year, usually three years. A rental is an agreement to drive the vehicle for a much shorter time period than one year.
Your description is of a rental.
There is a difference between leasing a vehicle and renting a vehicle. You specifically used the word "rent", and you say you only rented it for a month. Therefore, it's rent and should be reported as such.
Assuming you're filing a SCH C, you'll report your vehicle rental expense for that one month in the Other Common Business Expenses section. There's a sub-section for Business Travel which includes rental car expenses.