As you have discovered, mobile phone GPS locations are arrived at by two methods. The first quick method uses triangulation between the known location of cell phone towers. This gives a position that is probably good enough to help when you are lost. The system is quick and generally gives results accurate to within a couple of hundred metres in areas of good cell phone coverage. Accuracy is greatly diminished if all the cell phone towers are off to one side of the phone, such as on a coastline.
The internal GPS module in the phone relies on a small patch antenna that looks for GPS satellite signals. The signals are so faint as to be below the ambient background radio noise levels. The receiver repeatedly layers any signals received over and over until useful information can be recovered. This is then used to derive an approximate location and an accurate time and date. Once that has been achieved the receiver can then look for specific satellites based upon an almanac that is downloaded along with the time and position data. Eventually enough data is recovered to provide an accurate position.
A GPS cold start with no position data, outdoors with a clear view of the sky, can take ten minutes to establish a location. A warm start where an approximate position and accurate time signals are available from the cellular network or recent GPS use may only take twenty seconds. Trying to get a decent GPS signal inside a building without assistance from cellular ground signals may be impossible.
The latest modules are orders of magnitude more sensitive than previous generations of GPS devices and may get a solid position lock from switch on within 30 seconds. Also the latest GPS modules allow position data to be received and decoded from more than one constellation of satellites, so the US, Russian, Chinese and EU systems may all be used. My experiments with a uBlox module give positions accurate to with 10 Cm and a circle of uncertainty of 60 Cm. Good enough to locate the desk in my office next to a window.