Turboprop, I guess you are both basically right
A turbine engine may be a jet engine or it may not in the case of a turboprop.
Jet aircraft use jet propulsion not propeller propulsion.
A jet gets significant thrust from exhaust gases, a turboprop gets almost none from exhaust gases, some turboprops actually are reversed intake in the rear pushing exhaust out the front like the PT6.
Turbofans are considered jet propulsion as the both the exhaust gas and fan gas are pushing though an exhaust orifice rather than the fan being in open air as is the case with a turbo-prop.