The "no pressure" mantra is just wrong and, I suspect, causes beginners some problems. "Let the weight of the razor do the work" is equally confusing. Unless you lay on your side as you shave, the weight of the razor will pull the razor to the floor, not your face. Obviously, you do need to use a bit of pressure to hold the razor to your face and follow the contures of your face. This is obvious and yet the beginner (and I was there about a year ago) is told in bold, all caps letters "NO PRESSURE". It's just confusing and leaves the beginner thinking "I can't do this. I don't understand this. I don't get it. I'm doing this wrong." And, most hazardously, "my problems, including my nicks and cuts, are the result of pressure and I must focus on no pressure." No. You DO need to apply some pressure, just not very much. So, just relax on the pressure issue, you beginners. It's a light touch, a little pressure, and you'll figure it out quickly and naturally.
The no-pressure-mantra comes largely, I suspect, from people converted from electric razors where you really do have to press the razor into your face to force the hairs through the screen so that the rotating blades (yes, electric razors use blades -- the hairs are not, somehow, cut with electricity, so all shaving really is blade shaving) can cut the hairs. To prevent nicks and cuts, the blades in many cartrige-style razors are either wrapped with thin wires or have little bumps on them. Similar to the electric, these have to be pressed into the face rather firmly to get the a really-close shave (they are literally not intended to give a really-close shave -- they compromise this to safety -- and so pressing them to your face hard to get that super-close shave is really a mis-use of these razors and those who insist on doing this should just switch to a razor designed to give ultra-smooth shaves with minor pressure, e.g. a DE razor.)