Strictly speaking, you are wrong, I'm afraid.

A "Word" is the basic unit of data handled by the architecture; in general, its basic register size.
Therefore the size of a "Word" is entirely implementation-dependent:

A 4-bit processor has a Word size of 4 bits;

An 8-bit processor has a Word size of 8 bits;

A 16-bit processor has a Word size of 16 bits;

A 24-bit processor has a Word size of 24 bits;
etc, etc,...

a double is two words

Yes, of course!

Of course, the PC started as a 16-bit architecture (8088/8086), so the word size for a PC started as 16 bits and, of course, a double-word was 32 bits.