Passanger, many thanks for your observations. For your information, an engine failure at take off (EFATO) is probably one of the most practised manoeuvres that we as airline pilots carry out when in the simulator every six months.
It is fairly obvious from the incident where you were a passenger that the failure occurred before the aircraft reached it's V1 decision speed. Any major malfunction before the V1 speed and the aircraft can stop before running out of runway. Had the failure occurred after V1, the crew would have continued the take off.
On most aircraft we carry out performance calculations which take into account the weight of the aircraft, any deferred defects on the aircraft, the atmospheric conditions that affect the performance including temperature, wind direction and the condition of the runway whether dry, wet or contaminated, the length of the runway and stopway and any obstacles in the departure flightpath. Based on those calculations a power setting is derived which takes into consideration a catastrophic failure of an engine at the most critical time but which will allow the aircraft to continue its take-off within the remaining distance of the runway available and clear all obstacles during its climb to the minimum safe altitude. Also, if a reduced thrust take off had been used and one engine failed, the other engine can be brought up to full power should it be needed.
Once the aircraft has reached a safe altitude the crew would have carried out various drills to secure the failed engine, analysed the situation and made plans to return safely to an airport for a single engine landing. An emergency is usually declared when an engine fails after take-off and that allows Air Traffic Control and other aircraft in the vicinity on the same frequency to be aware of the situation and the aircraft will usually be give all necessary assistance to remain clear of other traffic.
I have never experience an engine failure in a jet and 99.9% of most airline pilots will never experience one for real. However, we are trained to handle the situation and no doubt there are few old hands on here who may have had the experience and will probably give you their accounts of the situation. Whilst an EFATO is not necessarily an emergency, it is handled as an "Abnormal Situation" and we have a very thick book of checklists which cover us for most "abnormal situaions", including an engine failure after take-off.
No doubt you were in good hands as it sounds as if the engine failure you experienced was handled "by the book".
