Engineers are used to dealing with these steps in bringing something new
to the market so they have a clear understanding, but as a woodturner I
confuse design with invention. I can confuse form with either invention
or design, but the difference is more often understood. Production and
promotion are usually distinct, but all of these steps depend on each
other. How do you see them?

Arch, you are not alone in your confusion. Let's start with engineers.
Engineering is basically applied mathematics. Aesthetic concepts like form,
color, contrast, composition, line and flow are not in their vocabulary. It is
Industrial Designers who create, ideate, and begin the process of making a new
product and taking it to market. Various disciplines might be employed along the
way including engineers, but not always.

Consider my story about a vase that an old time turner imagined...
His imagination in conceiving this new vase was invention.

Nope. What he did was called 'design' or 'craft.' It was not invention.
Invention is the creation of a new material, process, or product with unique
feature not yet produced. A popcorn bowl cum vase is at best, a 'object de
art.'

He bent some tool tips and bravely hollowed out the vase's insides
helped by a device to contain and control the tools.

Now, if his device was new, novel and did something other tool holders did not
do, then THIS is an invention!

Invention is a creative process and not confined to any particular profession.
Engineers are not more likely to invent something than designers, physicians,
housepainters, or clerks.

Dan