Hi Wookey
What you need to bear in mind is that the limiting performance (i.e. what sets the maximum weight that can be lifted off a given runway at a given temperature) is set by what an aircraft can achieve with an engine inoperative, i.e. after a failure at the most critical point on that runway.
The airworthiness authorities define a minimum climb gradient that must be achieved after this failure, and it is not very much! This is a major factor in the manufacturers ability to sell the aircraft and it may be (I don't know) that the 737 had a wider performance envelope (e.g. could operate at maximum weight) off shorter runways and at higher temperatures than the Trident which was after all "optimised" to the BA network, so wouldn't often be exposed to temperatures above about 30deg.
However, fortunately for us all, such failures are extremely rare - failures themselves are rare and for them to happen at the worst point is even rarer. What this means is that on a NORMAL takeoff without a failure, the aircraft actually has the power of one additional engine compared to that needed to make the minimum climb gradient. That translates of course into a much steeper climb since the speed stays the same.
So in the case of a three engined aircraft, it needs 2 engines to make the minimum, but normally has 3, i.e. 50% more power than the minimum. In the case of a twin, it can make the minimum gradient on only one engine, but normally has 2, i.e. twice what is needed. So a twin on a normal takeoff will always climb faster than a three engined aircraft, which in turn will always climb faster than a 4-engined one that only has 33% more than needed. (Not sure how the 1-11 fits into your picture though, I never flew it and it may have not been anywhere near its performance limits).