Airbus has optional fits for either EGPWS (Honeywell) or T2CAS (Thales). There should not be any major difference if both systems are certificated to the same standard (TSO-C151a).
The principle is that these systems provide boundaries for safe flight with an alerting (amber) awareness and a warning (red) commands. Amber level voice alerts are associated with amber lights, Red levels with a red light and where enabled, terrain-map pop-up depending on system type.
The red warning level requires executive action and is to pull-up wings level to establish the best rate of climb. Wings level has been subject to great debate, which in the early days of EGPWS involved all manufacturers, and the manoeuvre had to correspond with the terrain database/map accuracy. It is likely that as systems have developed, particularly with T2CAS, that in some circumstances a turn might be allowed, but always remember it is terrain (obstacles) that have to be avoided.
Amber cautionary alerting covers many scenarios and generally provides additional time for manoeuvre (but don’t depend on specific time values), - quickly evaluate the situation and adjust. Thus procedure wording can involve flight-path adjustment, climb, GA, turn, or change of configuration.
If a TERRAIN alert is given during an approach then a GA would be the better course of action as the EGPWS and procedures are designed such that alerts should not occur; if they do something is wrong and it is better to establish what at a safe altitude.
TERRAIN alerts at other times might be quickly resolved with flight path adjustment, but don't prejudge any situation assesment.
Other alerts during the approach such as Glide Slope or those involving Flap/Gear, might be subject to operator procedures, but in general the better option is to climb to provide time for evaluation.
Never try to second guess the situation; EGPWS reliability is now significantly better than the rate of human error – EGPWS is an error detector, yours or someone else’s.