Three headings. Three ways of perceiving the voice. Three terms which, together, confound the modern voice teacher. Why? They say too little and mean too much.
Let's take the last one. What is 'placement' after all? The voice doesn't go to a place, the scientifically-oriented teacher cries. Of course it doesn't. The vocal tract is the only resonator. But try telling that to the student who has been taught to obtain a ringing tone and is asked 'where' that tone is heard. "I hear it here," they say, as the hand rises to the level of the face. "I hear it out there." The hand is still at the level of the eyes, and gestures about eighteen inches away from the face. "I hear it here," as the hand points to the middle of the head. Should this information be ignored because the facial cavities do not resonate? Oh, that would be smart. So smart as to confuse facts with actual auditory sensation. And the latter is a reality even if little understood. It's also the chief means by which the student 'knows' what he/she is doing.
How to understand it? Science helps us here, ironically enough. But let's keep in mind that explaining the science to the student doesn't help them listen to the sensation. They need to keep the awareness of it in their mind regardless. Here's one study that was published in 2003 by Vurma & Ross, members of the Estonian Academy of Music.
Singing teachers sometimes characterize voice quality in terms of 'forward' and 'backward placement'. In view of traditional knowledge about voice production, it is hard to explain any possible acoustic or articulatory differences between the voices so 'placed'. We have synthesized a number of three-tone melodic excerpts performed by the singing voice. Formant frequencies, and the level and frequency of the singer's formant were varied across the stimuli. Results of a listening test show that the stimuli which were perceived as 'placed forward', correlated not only with higher frequencies of the first and second formants, but also with the higher frequency and level of the singer's formant.
What does this mean practically speaking? It means that the Old School teaching of taking all vowels from a highly resonant /i/ vowel—which is the most resonant vowel—results in the sensation of forward placement. How does one obtain a highly resonant /i/? Well, the vocal tract has to be lengthened for one thing. How is this best achieved? By the teacher modeling a deep, resonant, clear tone, one that isn't too loud—from god's mouth to the student's ear—which is how singing has been taught since the 18th century. Sure, the teacher can tell the student to lower their larynx manually, but this doesn't achieve the desired tone because mere physical manipulation doesn't engage the students ear—a very necessary thing. However, get the student to make the desired tone but engaging their ear, and then ask them how it feels? That's a very different matter. The kinesthetically aware student will report a slightly lowered larynx. Yes, the soft palate will be felt to rise too, but this isn't always so easily noticed by the beginning student, who needs to drive the car around the block awhile before matters become clear.
Voice placement. Call it an auditory sensation, a combination of bone and air conduction, the singer's formant, ring, ping, point, chiaroscuro tone etc. Whatever the terminology, it's the teacher's job to help the student obtain it. To do it, the teacher has to engage the student's ear. (Did you catch the line in the quote above? "Results of a listening test...") The student has to be taught how to listen and what to listen for, which is altogether different than thinking about scientific facts while singing. The latter doesn't help the student one iota.
Manuel García - the father of modern voice science - didn't teach singing using scientific terminology because he didn't want to confuse his students, leaving that knowledge for the curious and those who intended to become teachers themselves. He did teach, however, his students to be aware of voice placement (click on the 'voice placement' label and find my post on Herman Klein & The Bel Canto for more information). The modern voice teacher who doesn't understand what this means will insist that the student not listen to what they are doing. And what will be the result? That teacher will rob that student of his/her audio-vocal control. Now. What's so smart about that?