Your provided reference is a non-scientific site. The author offers no real credentials. They are a poor source of information. Anyone can make a .com or .org site and say anything they want; avoid them as primary sources.

Here, for instance, is a key bulletin that handset manufacturers are required to follow in order to make sure their handset receivers implement the 2nd-order relativistic corrections needed to adjust for non-standard satellite orbits, which affect GPS signals beyond the frequency corrections built into the satellites to handle the bulk of the time dilation effects:

http://www.gps.gov/technical/icwg/IS-GPS-200E.pdf

As you can see by searching through the document, there is extensive mathematics to implement in software to make the final corrections and achieve full possible accuracy.

It's unfortunate the one engineer you spoke with didn't know this. putting you on the wrong path.

Newton + Maxwell are good, but insufficient to understand the universe.