Open Throat
It's a term that has a long history, like that of 'voice placement' and 'support'. Of course, all three have to be experienced to be understood. Why? Because they refer to physical and auditory sensations rather than actual physical events. Confused by that statement? You aren't the only one. I remember being fascinated years ago when I learned that there is more physical space in the pharynx made by an /i/ vowel than /a/.
In the confusion that can arise in obtaining the experience that a term implies, it should be understood that mechanically 'making space' to experience an 'open throat' is as counterproductive as pushing or pulling the abdomen this way and that to acquire 'support', or 'sending' the vibration of the voice to the hard palate in order to obtain 'voice placement'.
Regarding 'open throat'. What did Old School pedagogues teach? That it was acquired through an /a/ vowel and exhibited a free and full tone. And what is a 'free' tone? One that gives both the listener and the singer the perception that the throat/pharynx is not obstructed in any way, specifically, that the tone is neither guttural or nasal. This is expressed in the Old School saying:
The Italian Singer Has No Throat!
This means, practically speaking, that the singer's audition of free & forward tone so monopolizes his/her attention that the throat doesn't even come into awareness.
Domenico Corri, a student of Nicola Porpora (who is considered the greatest vocal maestro of all time) recorded in 1810 in his book The Singer's Preceptor...
...Porpora, whose decided opinion it was, that solfeggio were not properly understood; the improvement of the voice he maintained is best acquired by sounding the letter A—the position of the mouth in uttering this letter being most favorable to produce a free and clear tone.
Manuel Garcia distilled this vocal wisdom into physiological terms in his groundbreaking work A Complete Treatise on the Art of Singing (part one: editions of 1841 and 1872 collated, edited and translated by Donald V. Paschke).
The Study of Tones
The Stroke of the Glottis [Coup de la Glotte]. Hold the body straight, quiet, upright [d'aplomb] on the two legs, removed from any point of support; open the mouth, not in the form of the oval O, but by letting the lower jaw fall away from the upper by its own weight, the corners of the mouth drawn back slightly, not quite to the point of the smile. This movement, which holds the lips softly pressed against the teeth, opens the mouth in the correct proportion and gives it an agreeable form. Hold the tongue relaxed and immobile (without lifting it either by its root or by its tip); finally, separate the base of the pillars and soften the entire throat. In this position, inhale slowly and for a long time. After you are thus prepared, and when the lungs are full or air, without stiffening either the phonator [1872: throat] or any part of the body, but calmly and easily attack the tones very distinctly with a light stroke of the glottis on a very clear [a] vowel. That [a] will be taken well at the bottom of the throat [1872: right at the glottis] in order that no obstacle may be opposed to the emission of the sound. In these conditions the tone should come out with ring and with roundness.
Lest the singer become involved in convoluted efforts to obtain the physical conformation above, it might helpful to know that Garcia's instructions are better understood if awareness of a vibrant and resonant tone is kept in mind rather than one's muscles. It's the difference between breathing through a tone and breathing to make a tone. In the first instance, the mind's eye is focused on the tone—not unlike how a tennis player's eye is kept on the ball. I like to think of the matter this way: tone first, mechanics second.
One thing is worth noting regarding Garcia's instructions: at no time does he tell the student to mechanically lift the soft palate to make space in the pharynx. This happens—it would seem—as a result of taking a deep breath and opening the mouth in the prescribed manner (that the larynx also descends slightly is a given). To think of it another way: when do you take a deep breath as a matter of course? When you have something important to say, and want to 'make it clear'.