The explanation is right, only your counting isn't.
Sol is the fifth note in C major: G.
The numbers start at "1" for the highest ocarina that Donati made, which is equivalent to our "soprano" C. Then follow ocarinas alternately in G and C, down to the contrabass C7.
The Budrio names with their fundamental notes, first in scientific pitch notation and then in Helmholtz pitch notation, and then in common modern names:
Do 1 = C6 = c''' = "Soprano C", Langley: "Mini"
Sol 2 = G5 = g'' = most often "Soprano G", STL: "Alto G", Langley: "Soprano"
Do 3 = C5 = c'' = most often "Alto C", STL: "Tenor C", Langley: "Alto"
Sol 4 = G4 =g' = most often "Alto G", STL has none, but if they had: "Baritone", Langley: "Tenor"
Do 5 = C4 = c' = most often "Bass C", STL: "Bass C", Zin: "Tenor", Langley: Bass
Sol 6 = G3 = g = I only know the Langley: Great Bass
Do 7 = C3 = c = most often "Contrabass", Langley: Mega Bass
I actually prefer the naming systems of STL's and Langley's ocarinas, because they follow a tradition that lasts for more than five centuries, and because they have an equivalent in the human voices, where a soprano sings one octave higher than the tenor, and an alto one octave higher than the baritone or bass.
Soprano C, Alto G, Tenor C, Baritone G, Bass C, Great Bass in G and Contra Bass in G would be a logical way to name the instruments.
Or alternatively:
Sopranino C, Soprano G, Alto C, Tenor G, Bass C, Great Bass in G and Contra Bass in G.
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