Recommended pressure always varies. With wider tires, you typically must run them lower, since a too-high pressure will cause the tire to blow off the rim.
If the tire is a stiffer material, this dissipates more energy. In addition, suppler tires deform over obstacles more readily, reducing bouncing and vibration.
More or less, your contact patch is pressure/weight, since the tire can't support much weight on its own. It's going to be almost the same no matter what (slick) tire you are running. Wider tires have less rolling resistance for the same pressure (to a point), since the tire deflects less to maintain the same contact patch. Two tires of the same width, one stiff and one supple, will have to bend in the same way, and have nearly identical contact patches. However, the supple one will dissipate less heat doing this.
This is not correct. Optimum speed comes are optimum pressure, which is usually quite a bit lower than most cyclists imagine. Going too high results in worse rolling resistance, AND worse handling AND worse comfort.
As a ballpack figure for a road bike, a 25c tire will usually have optimum rolling resistance at around 90-100psi for typical male rider mass and road surfaces.
And the faster, better handling tires also have less rolling resistance, but less durability.
We're living in interesting times for this "conventional" wisdom! One school of thought argues that while there might be a theoretical relationship between high pressure and low rolling resistance, but the real-world effect has been exaggerated by poor testing methodology, which usually involves a circular drum as a substitute for a flat road. And that the benefits of achieving low rolling resistance are offset by the losses due to increased vibration.
Of course, this school of thought is largely the product of one man, Jan Heine of Compass Bicycles, but it's interesting to read and consider. Personally, I made the jump from 23mm road tires at 120psi to 48mm slick tires at ~40psi this year, and I have to say, my average speed didn't take a hit at all on the road and am much happier and more comfortable.
https://janheine.wordpress.com/2018/05/02/testing-tires-isnt...
https://janheine.wordpress.com/2016/03/09/tire-pressure-take...
However,even though they feel faster, they’re actually slower. That extra vibration is coming from vertical deflection that a more supple (lower presure) tyre would allow to stay as forward momentum.
Personally I keep “25mm” (GP4Ks always seem to measure closer to 27mm for me) somewhere between 70-80 psi but then I’m the sort of madman that thinks I can feel the difference with latex inner tubes.