Yes you are.
BHBA is intimately related to acetoacetate. They are just one hydrogen atom different and metabolically there is a lot of conversion between the two. BHBA breaks down into AcAc and they both break down into Acetone.
Both BHBA and AcAc have an acetone molecule in them. The statement you quoted isn’t meant to exclude BHBA. It’s just that AcAc is more direct. Think of it as a 1-2-3 and 3-2-1 process. Acetone <-> AcAc <-> BHBA.
You can also see this in the diagram you posted, that AcAc has 2 Acetone molecules in it linked by a carbon bond, and BHBA has substituted one Oxygen atom for a Hydrogen instead. So the chemical reaction description of the breakdown is merely described differently “decarboxylated”.