As you already know, GPS satellites are not geostationary, the system is composed of 24 satellites Navstar placed in an orbit inclined ~ 55°at the altitude of 20,184 km (12,540 miles).
Because GPS satellites are not geostationary then always moving, our GPS receivers store in memory the information from the last locations that it was on satellite signal, which explains why satellite acquisition may take longer if you have not used your GPS receiver since more than 1 week.
Also Monitor Stations upload new, corrected data to each satellite every four hours. This data includes a corrections on the exact time and position of that and other GPS satellites in orbit. An update of the satellite’s position can be determined by performing a GPS measurement to a ground antenna of which the exact location is known. The Monitor Stations are located near the equator to reduce the ionospheric effects.
The geostationary orbit (also called Geosynchronous) is very different, it is circular orbit roughly 36,000 km (22,369 milles) above the equator. The satellites positioned in a geostationary orbit provides capacity for Video (TV channels), Radios, Data and Multi-Usage Services, to speak only of civilian satellites.
Only China is going to have 5 geostationary satellites to provide positioning system with its Compass system that will also include 30 non-geostationary satellites.
Thank you for posting about the "Old Transit Sat System", this remind me my preparation for the boating license exam in the 90'! Transit was the first satellite system, launched by the USA in the 60' for the US Navy. The system, based on the principle of the Doppler effect, was revolutionary. (Radar guns, for example use the Doppler effect to measure the speed of vehicles or other objects such as baseballs).
Transit was six satellites on six polar orbits, altitude: 960 km (596) miles, orbiting the earth allowed ships to fix their position on the seas once every hour. Transit could guarantee a successful measurement within 110 minutes at the equator, as long as a satellite was in range of the receiver. The greater the latitude, the more satellites became visible. For example, at 80° latitude N, the average fix time was only 30 minutes. (If my memory serves me well, correct me if I'm wrong here).
It was mostly when the USSR shot down a Korean passenger jet - flight 007 - in 1983 that the Reagan Administration had the incentive to open up GPS for civilian applications which was done on May 2, 2000, by the Clinton Administration.
After 17 years and numerous setbacks and budget boosts, Europe's Galileo satellite navigation system has gone live. At this point, 18 of the planned 30 satellites are already in orbit. European Union launched Galileo, a planned satellite navigation system, intended as a European alternative to the US GPS. It is currently under consideration by the European Union and the European Space Agency. The system is intended to be primarily for civil use, unlike the US system then Galileo will increase geo-localization precision tenfold. Besides the European Space Agency member states, numerous other countries are involved in Galileo in one way or another. Among them are India, Israel, Morocco, Saudi Arabia, South Korea and Ukraine.
A modernization program of the US GPS system would also contain standardized features that will allow GPS III and Galileo systems to inter-operate, allowing receivers to be developed to utilize GPS and Galileo together to create an even more precise positioning.