I don't understand why these clauses exist.

I'm a contractor, with a company that previously employed me full time.

When I was full time, the employment contract was pretty draconian. It said, pretty much, "Everything you do is owned by us."

Now, my employment contract says, "No plagiarism".

That's it... "Write your own code".

So, let's see. I'm paid more, have less stress, and my employment contract lets me tinker on my own time with no restrictions. All so the company can sack me with 1 month notice (something that they are inclined to do to full timers anyways). Where do I sign?

So, how about:

"If it is done on work time, or with work resources, $COMPANY owns the copyright and work. Otherwise, it is owned by the employee. However, $COMPANY expects the employee to not compete with $COMPANY during the time of their employment. If the employee produces a new idea that is related to our current field of effort, we expect first right of refusal prior to the employee working on the invention. If we refuse, that refusal is non-reversible."