The short answer is that the relationship is not a simple one but depends on many other factors.
The example that you give of a cold front has more to do with the nature of the shape of isobars around a Low with fronts trailing from it than a direct relationship between temperature and pressure. The low will tend to move parallel to the warm sector isobars, so as the cold front passes, the pressure will rise.
If you heat a column of air, causing it to rise, it will tend to expand upwards, thus creating a core of higher pressure at the upper levels. This causes it to diverge a little at the upper levels as the, in effect, drifts out from the high pressure core. In turn, that means less air in the column, and so less pressure at low level. Thus, all else being equal, a column of hot air will tend to be associated with low pressure close to the surface, and high pressure in the upper levels.
Put another way, the rate of decrease of pressure with height is inversely proportional to the temperature.
Does that help to reconcile your apparently contradictory "theories"?