To be honest you are confused in your opening post earlier than you think. Low pressure areas (depressions) give rise to both hot and cold fronts. High pressure areas (usually) don't have fronts.
Fronts are just divisions between warm and cold air. Whether they are called warm or cold fronts depends on their movement and whether an observer on the ground experiences a rise or fall in temperature as they pass.
Bear in mind that a lot of the words we use to name things in the weather are based on poorer understanding than we now have - names given to things by Victorian gentlemen enthusiasts who had no satellite pics or computer modelling. The person (this guy
Jacob Bjerknes - Wikipedia, the free encyclopedia ) who first called the division between airmasses a "front" did so because he thought it reminded him of the armies facing each other across the "front" of the trenches in France in the First World War.
Its too complex to explain properly without diagrams, so best not to try. If you want to explore meteorology further, and its a fascinating subject because even the professionals don't understand as much as they'd like to, I think you should also read this