Unfortunately, the source you cite doesn't give any references, but a similar description is also given in Avoiding the Tooth-Pullers: Dental Hygiene, Eighteenth-Century Style, which cites Dr. Johnson’s London by Liza Picard. She cites Antique Dental Instruments by Elizabeth Binnion, who in turn cites the French barber surgeon Ambroise Paré. Unfortunately, very little of the text can be seen but Binnion says Pare (apparently with reference to a specific instrument)
advised that the patient be seated on a low seat with his head between
the operator's knees, the gums scarified, the tooth shaken and forced
with an elevator, the forceps used as a last resort..
Pare (circa 1510 - 90) was primarily a surgeon with much experience on the battlefield. The article you cite does insert 'quite likely' (for 'no stool') and 'may' (for 'sit on the floor') but it does perhaps give a slightly misleading impression if contemporary visual evidence is anything to go by.
Looking at images from the Medieval and Early Modern periods, 'dentists' (or rather barber-surgeons, tooth pullers or blacksmiths) are shown in a number of different positions vis-a-vis the patient, and there are various descriptions as well. Based on Medieval and Early Modern images, the most common methods seem to be either (1) with the 'dentist' in front of the patient, sometimes using his knee or foot on the patient's chest or (2) from behind, with the patient's head tilted back and often held with the 'dentist's' free hand / arm or (3) with a third person holding the patient's head from behind as the 'dentist' pulls the tooth while facing the patient. (see images below)
Teeth were also sometimes pulled by tying a piece of string to the offending tooth, with a drum beating in the background to distract the patient. Various other positions / methods were also used, though many were undoubtedly not common. Tooth pullers in France were often exhibitionists who performed outdoors for an audience with the intention of impressing spectators with various 'clever' ways of extracting teeth:
One famous tooth-puller, for example, was known for his trick of
extracting a tooth with one hand while firing a pistol in the air with
the other, and with his head in a sack. Others performed the act of
extraction while seated on a horse
Pierre Fauchard (1678-1761), known as the 'Father of Modern Dentistry' was apparently the first to champion the need of comfortable seating during operations, but we're still some way from Josiah Flagg's first dental chair (1790).
Below are a selection of images showing teeth being pulled, and more can be found by clicking on the links in this post. Keep in mind that some of the images are caricatures, though I've left out the more 'exaggerated ones.
From a Persian manuscript by Abu ʾl-Qasim Firdowsi Tusi (930-1020). Source: Witnesses in the dental scenes in old paintings
"a dentist with silver forceps and a necklace of large teeth, taken from the Omne Bonum, published in the 14th century." Source: A history of dentistry – in pictures
"The Tooth Puller (detail), by Honthorst Van Gerrit (1590-1656). Photo © RMN – Grand Palais (Musée du Louvre) / Stéphane Maréchalle." Source: The rise and fall of the smile in 18th-century Paris A study in dentistry
By Dutch painter Lambert Doomer (1623-1700). Source: Witnesses in the dental scenes in old paintings
The Country Tooth-Drawer (1784). Image source: British Museum