why did they not ask for fire services in that first call?
You're a long way forward of the engines in the 777 and they aren't visible from the cockpit. The fire loops give indication of a fire within the engine housing. This warning can also be a 'hot gas leak' coming from either a blown seal or a crack in the casing.
The fire warning will cause the 'Stop' call to be made then there is a pause for all three pilots to confirm the indications and identify the potential cause. Once the cause has been established then the appropriate call can be made.
The delay is to ascertain exactly the severity of the situation as putting your passengers out of the aircraft, down the slides onto an active, busy airfield isn't a decision that any Captain would take lightly. Once the initial 'alert' call has been made and the fire verified then the second call would be made.
Simple answer never rush!