First of all, let's clarify here what we are talking about. These suggestions are talking about how to tune a traditional non-locking tremolo that is floating, meaning the back of the tremolo is raised off of the body, allowing both upward and downward pitch changes. It is possible to tighten the tremolo springs enough to bring the back of the tremolo down to rest on the body, and the tuning method for that setup is basically like tuning a hardtail non-tremolo guitar.
But for a working, floating tremolo, tuning can be frustrating if you don't have a method to the madness! Tuning one string effects the tuning on all the other strings so you have to find a balance where all the strings are in tune and the tension is equalized with each string on both sides of the nut. If, for example, one of your high strings has more tension behind the nut, as soon as you bend a string it goes flat because it has compensated.
So the process I have found that works the best is to tune several times across the strings until all the strings are in tune or very close, dip the tremolo down and tune again. Sometimes you need to do that a few times to get it just right but if your tremolo is set up correctly and the strings can move freely in the nut it should work.
Traditional tremolos took a giant leap forward with the introduction of locking staggered tuners such as Sperzel and Schaller. Because the posts are staggered, no string tree is needed and that eliminates one more source of "hangup" as well as some tension across the nut, making it come back to pitch more accurately.