Back in the dark ages when I was trying to develop a nice vibrato I used long tones to work on it, but I wouldn't confuse this with the practice of long tones for the purpose of tone development and embouchure control. Practicing long tones with no vibrato is important. The fact that you use "long tones" when practicing vibrato isn't really long tone practice...it's vibrato practice...a different exercise entirely (in my opinion). The two do tend to merge when you work on delayed vibrato. When it comes to personalizing your vibrato, you're mostly concerned with "how wide", "how fast", whether or not it's delayed or immediate, and whether or not you use it all the time or just "here and there". You should also be able to control your vibrato and change it at will to suit the style of music rather than using the same style of vibrato on everything you play.
Whether for better or worse, I had to figure out vibrato on my own. I never had anyone teach me how to control it. After much frustration about not knowing how to do it, I came up with my own exercises using a metronome...playing long tones on a single note...subdividing the beats into rhythmic vibrations based on eighth notes, triplets, and sixteenth notes...gradually increasing the tempo of the exercises over time as my ability to control the vibrations progressed. Of course this isn't how you use vibrato in the real world. Vibrato isn't a precise rhythmic subdivision of beats, but what those exercises did was to help me develop the feel for how to create vibrato and how to control the speed and depth of it at will. The hard part after developing the control was forgetting about the specific subdivisions I had been practicing and learning how to maintain the same kind of control without it being so beat-specific. In other words, playing with a more natural voice-style vibrato, and learning how to change it up if and when necessary or desired.
The short answer to your "related question"...I don't use vibrato when practicing long tones...but I do use long tones when practicing vibrato.