What's so special about Chassaignac tubercle?
The reason massaging the carotid artery can relieve symptoms of supraventricular tachycardia is because by massaging the artery, as you point our yourself, you artificially increase BP within the carotid artery, this activates a reflex (via the glossopharyngeal nerve) that lowers bloodpressure by reducing the heart rate via the vagus nerve, by increasing the refractory period of the AV-node, so the heart rate drops.
It should be noted though, that it can be a bit dangerous to play with these reflexes in some patients, especially because massaging the carotid arteries can lead to an embolism in older individuals with atherosclerosis, or in some patients, it can lead to asystole or ventricular fibrillation (i.e. cardiac arrest).
It's not really about which vertebra is at the level you are massaging, rather it's about being near the carotid sinus, which is where the baroreceptors are located, and that might vary between individuals and how you position the head etc.