AMT can cover a few different systems. Modern versions typically use a dual clutch system which pre-selects the next gear. The actual driving experience will depend on the control system and it's not uncommon to have selectable modes which work in concert with engine mapping, traction control, stability control ABS etc.
In many cases they will still have a manual clutch, either as a hand paddle or foot pedal and may also have a neutral button as well as an automated anti-stall system. The clutch is not required for normal gear changes but may be used for pulling away from a standstill. Equally AMTs can emulate traditional type automatic transmissions and just use throttle and brake with no driver input to gear change or clutch operation at all.
As an aside, there is quite a lot of technical information on this subject in the context of F1 gear boxes. Until recently F1 cars had a dual paddle clutch system with one paddle used to blend in the clutch from a preset bite point and one which was more of an on/off switch with respect to the initial preset bite point. The idea being to improve reaction time to the start lights as the first one is was released immediately in reaction to the lights and the second was released gradually to give fine control of torque delivery.
So in summary a modern semi-automatic gearbox has quite a lot of potential parameters available which can be adjusted to affect the driving experience which is determined by a combination of user settings and the software which controls the whole system. So in a sophisticated version you might have settings on a spectrum between convenience and ease of driving and maximum performance with a range of settings in between as well as modes for specific conditions and to maximise fuel economy.