Ballo del Fiore for two is unusual in that the instructions contain the phrase "principiando ogni cosa col piè destro" - beginning everything with the right foot. This is the only mention of which foot to use first.

In Ballo del Fiore for 3, found on the same page of Caroso's "Il Ballarino", the instruction to start some part of the dance with the left foot occurs 4 times and in Ballo del Fiore for 4 or 5 it occurs twice, including "principiando tutti i Moti in questo Ballo col  piè sinistro;" - beginning all the movements in this Ballo with the left foot.
In no place in either dance does any section begin with the right foot.

Since starting with the left foot is the rule in the overwhelming majority of Caroso's dances, one must suspect that the instruction to start on the right is a misprint.

However I don't really think it would be worth the effort to try to change the way the majority of people, in Lochac at least, do this very well known dance.

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That said, there is a section I would reconstruct differently in the light of information gleaned from the other two versions of the dance - the section where Caroso tells the dancers to do four Seguiti ordinarii, two turning to the left and two forward.

In the commonly used reconstruction (see Del's Dance Book) the dancers, who begin facing one another, go forward passing left shoulders and then curve around to the left to face one another again. This reverses the order of the original instructions about direction - 2 turning _then_ two  forward - presumably in order to have the dancers facing one another again.

They are next required to do two Continenzi (the 'singles' in the reconstruction linked above) and a Riverenza, followed by 4 Seguiti flankingly and two more Continenzi and a Riverenza, after which the gentleman gives the lady the flower.
Certainly they must have been facing one another at the beginning of this sequence in order to be able to present the flower at the end of it, but I don't believe that it is necessary to ignore the order of Caroso's instructions in order to have them do so.

In Ballo del Fiore for 4 or 5 there is a section where the dancers pass one another and go in opposite directions using 4 Seguiti (Seguiti semidoppii ornati). It is obvious that they then turn to face one another - the next section requires them to do much the same steps as described in the section of 'for Two' above, specifically approaching with four Seguiti done flankingly. This is preceded by two Continenzi and a Riverenza and followed by two Passi puntati, a Riverenza and the presenting of the flower.

There is, however, no instruction to turn around at the end of the first four Seguiti. As the turn must happen I have the dancers turn in the last of the Seguiti.

Since the relevant section in Ballo del Fiore for Two has the forward Seguiti after the turning ones and the lack of instruction to turn from Caroso is no bar to there being a turn, at least when the next direction of movement is obvious, I would say that in Ballo del Fiore for Two the dancers should also turn in the last of the two forward Seguiti.

The two preceding Seguiti, which are described by Caroso as turning, would most likely an individual circle turning to the left, as in my reconstruction of Ballo del Fiore for three, but the dancers could instead circle around a common point, most likely changing sides.