I am wondering wether I should match the recommended PSI of both my tires (equalized) or if I should have the front tire slightly more deflated than the back or vice versa (offset). Are there occasions where front and back tire pressure is slightly and preferably modified, such as a pro cyclist preparing a road bike for a time trial in comparison to a mountain bike for difficult terrain.
Because there's (usually) more weight on your rear wheel than your front, you want to run your rear tire at higher pressure to get an equal amount of "drop" (tire deformation) on both of them. One theory is that you should aim for 15% vertical deformation. Here's a calculator that can help.
Tires usually have a inflation pressure range, or maximum pressure specification rather than ab absolute recommended pressure. The pressure you should be using depends on how much you weigh, the surfaces you are riding on and your preference for ride comfort.
Unfortunately there is no way to definitely determine what you tire optimum tire pressure should be, but you can experiment with different pressures.
Similarly there does not seem to be any definite guidance for different front and rear pressure other than the front should be a little lower because most riders have a rear weight distribution bias.
Personally, I ride a drop bar bike with a moderately relaxed position so I run the front about 10-15% lower than the rear.
I always put near maximum pressure for the rear tire, because it will support most of the weight of me and my bike. For the front tire, I put slightly less pressure. This will improve handling because less pressure will make the tire can dampening the rough terrain.