Yup and I would like to add.
In this case x86 refers to a 32-bit instruction set that can be translated by the x86-64 instruction set (the software), for the instruction set architecture (an intermediary between hardware and software), which in turn translates it for any given 64-bit CPU microarchitecture (the hardware).
When talking CPU architecture it is 8, 12, 16, 32, or 64-bit and as you correctly pointed out the x86 refers to a family of 32-bit processors. While I understood your last sentence and consider it to be correct, allow me to rephrase the last part as:
They are 64-bit CPUs capable of running the x86-64 instruction set.
Just to avoid confusion.