The public form is best known from sculpture. Roman relief panels show the three animals in procession, sometimes with attendants and musicians, moving towards an altar where a magistrate stands with his head covered by a fold of his toga, which is the Roman manner of sacrificing and the visible difference from the Greek, who sacrificed bare-headed and garlanded. The scenes are common enough on public monuments that the suovetaurilia became a standard visual shorthand for the state performing its duties properly, which is a use of a rite rather than a description of one.
The army version purified the soldiers as a body before a campaign or at a fixed point in the year. The census version closed the enumeration of the citizens: having counted and registered the Roman people, the censor purified them, and the counting and the cleansing formed a single administrative and religious act. It is worth noticing what this implies about Roman thinking. Impurity here is not moral guilt and not contact with a corpse; it is whatever has accumulated on a collective over a period of time, and the remedy is applied to the collective as such, on a schedule, by an official.
The private and public forms are the same rite at different scales, and the sources do not treat the farmer's version as a folk imitation of the state's. If anything the reverse is implied, since the agricultural handbook records the older formula. What the rite really required was a boundary, three animals and somebody entitled to speak the words.