Power saving mode (PSM) is what gives long battery life. The device stays registered but switches the radio off completely and cannot be reached until it wakes. Two timers are negotiated with the network: the active timer T3324, the time the device stays reachable after finishing a transfer, and the periodic tracking area update timer T3412, the longest it may sleep before it must check in. They are requested with AT+CPSMS, and the network may grant different values from the ones you asked for, so read back what was granted.
eDRX is the middle ground: the device stays reachable but listens for paging only every so often, from seconds up to many minutes, requested with AT+CEDRXS. For a tracker that only sends on its own schedule, PSM fits. If you need to send it commands with a delay of a minute or so, use eDRX.