If the watch is worn on the wrist, the steps are therefore counted by how many times we swing our arm and have nothing to do with the number of steps we take. So would it be more accurate to wear the watch on our ankle?

Yes, but it's not always practical.

When I walk the golf course pushing a golf cart in front of me, obviously my arms are not moving, therefore my watch shows that I went about 1,000 steps when the golf course is actually 7 miles. Same thing when I ride my stationary bike, holding the handlebars, the watch does not register any steps. In those two circumstances would it get an accurate count if worn on the ankle?

For biking the step counter is turned off. Normally you don't take any steps during biking. My guess it is turned off since vibrations could otherwise be logged as steps. 

It will work for the golf cart. 

Get a cheapo VS4 or similar to wear on your ankle and wear the activity watch on your wrist. Of course both devices must be registered to your account. Just keep the activity watch as the primary device.

4411990 said:

Yes the step counter is turned off in the cycling activity, BUT if you wear a Garmin watch on your ankle while you ride, you will see your step count increase in the day's total.

No, you won't. At least not with current Garmin models, and when using any cycling type activity (and a few others - see the  document Why am I Not Gaining Steps During Certain Activities | Garmin Customer Support for the full list). A few years ago daily steps were counted even during cycling activities, but that was intentionally disabled by Garmin. It may be still possible with some very old models where the firmware was not updated in the last few years.