Is there an abundance threshold of a 'synthetic' element for it to be considered natural?
Is there a natural abundance threshold of a 'synthetic' element for it
  to be considered natural?


$\mathrm{&gt; 0}$

In other words, any element that can be detected outside of the laboratory (e.g. in the natural environment) is a naturally occurring element.  This would include the 80 stable elements along with the 18 radioactive elements with very long half-lives, and the shorter-lived radioactive elements produced upon their decay.  

Therefore there are 98 naturally occurring elements; all of the others are synthetic, artificial, man-made, whatever you choose to call them.

This Wikipedia article provides nice background, especially the following sections:


the introductory section, especially the last paragraph therein
Occurrence and origin on Earth
List of the 118 known chemical elements - Occurrence