When is a double g not a double g? It would seem to depend on whom we talk to. Theorists and violists would maintain that what we call a “double C” (c4), is in fact a b-flat 3, or in other words, a “high Bb.” This is because notes names are based on the lowest note of a scale starting on c, be it Sub-Contra C or whatever, and all the ascending notes belong to the same species. Hence, our double g must be higher, not lower, than a double c.
Trumpet players are not violists, however, and those that are theorists are thankfully first and foremost trumpet players.
In trumpet-speak of ages ago our pedal c was called “Basso.” Our low g was called “Vulgano.” Middle c, e, and g belonged to the “Principale” player. The “Clarini” got to spend most their time between c and high c, sometimes dipping down to what was for them a “low” g (even as low as the Vulgano g, as in the Christmas Oratorio or up to a double g in Michael Haydn). A notated c an octave above high c is rightfully called a double c, and for trumpet players the g below that may be rightfully considered a double g. That does not, however, excuse us for playing without taste (unless attacking violists).