By air moving from high pressure area to low pressure area means that it's moving from colder area to a warmer area, are they both not contradictory? I might have understood something in a wrong way, pls help.
You are correct that air generally moves along the surface from high pressure towards low pressure (not directly, but deflected due to the coriolis effect).
However, this does not imply that it is moving from a colder area to a warmer area. And, there is nothing contradictory in a cool air mass moving horizontally to displace a warm air mass.
Extratropical cyclones (a kind of low pressure system found in mid latitudes) are actually related to the movement of a cold front and a warm front, as part of a frontal wave. They drive cold and warm air masses to move.
Your claim that "air moves from warm object to a cold object" might be the root of your misunderstanding. That claim isn't true. It is true that heat will move from a warmer object to a cooler object through conduction and radiation. But that is not related to whether the air itself will move. A local effect called a "sea breeze" demonstrates the opposite scenario. A hot beach creates a warm, rising column of air (and a local area of low pressure), bringing in cooler air from the ocean to replace it.
You have conflated the transfer of heat with the movement of air and air masses.