A speedtest app tells you the speed between your phone and the speed test server - that's the network you're checking. If the server is close to your carrier or internet provider (mobile data or wifi, whichever you're using at the time), that's your speed to your provider. If you choose a server halfway around the world, you'll see that "your" speed has dropped - that's because "network" speed is measured between two points, you and some server - and your speed to some slow computer connected to the internet through 10 miles of wire hung on trees is going to be slow, but if that's the only server for that speed test in that part of the world, they might accept it on their service.
(Up and down speeds are the speeds that are being measured. Your provider normally limits your up speed to less than your down speed, because what most people do up, like what they're typing on a keyboard, is slow. My current upload speed [I just checked] is about 1/4 my download speed.)