Hexagonal circle packings in the plane

I have been reading the paper Spiral hexagonal circle packings in the plane (Alan F. Beardon, Tomasz Dubejko and Kenneth Stephenson, Geometriae Dedicata Volume 49, Issue 1, pp 39-70), which proves that “these ’coherent’ [Doyle] spirals, together with the regular hexagonal packing, give all possible hexagonal circle packings in the plane”.

On the obvious reading, this claim is straightforwardly false. For example there is a double spiral hexagonal packing obtained as a Möbius transformation of the Doyle spiral, as illustrated here by Francesco De Comité:

Double spiral

Am I interpreting the claim incorrectly? How ought it to be understood?

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