Is there a difference in symbolic meaning between the two representations? Is there a point at which it definitively became one-headed?
One-headed and two-headed versions were, as you noted, seen throughout history. I know of no significance for the variants.
As for "symbolic meaning": in general, symbolism was a focus of heraldic treatises, for example, but it appears to have generally been invented willy nilly with little consistency, so it had little real basis. Sometimes there were straightforward meanings for elements of a coat of arms, but the ones I know of are
(1) showing a close familial relationship to another by using their arms with marks of difference ("cadency").
(2) references to someone else's arms, like using "ermine" in coats of arms in Brittany, or in Italy, taking the arms of the Empire or of France on a chief (horizontal area at the top of the shield) as "arms of allegiance".
(3) Puns on the family name, like the head of the Talbots having "talbots" (a breed of dog) on his arms.
The Eagle as a heraldic device is rooted in the Roman Aquila standard carried by the Roman Legions. It was adopted in Germany as the emblem of the Holy Roman Empire, an effort along with the name itself to connect it to the legacy of the Romans.