Wat is an T1 and T2
The A1 and A2 terminals of a triac are NOT interchangeable.
An image search reveald they're somtimes also called "MT1" and "MT2", maybe other names too. But still, they are not interchangable.
In the schematic with the MOC3021 and the BT136 you only have a resistor to limit the trigger current.
In the schematic with the MOC3061 and the BTA41 you also have a resistor to lower the sensitivity of the triac.
Why do you think (assume?) these different components would work in the same schematic?
The values of the snubber network are also different between these schematics.
When looking at some pictures I saw this:
http://www.penguintutor.com/It shows clear differences between the MT1 and MT2 terminal.
Ah, the resistor...
Whether you need (or are wise to) use the extra resistor depends on a lot of things.
Simple answer is, sometimes it's needed, and it almost never hurts to put it in.
Main factors are the leakage of the Opto Triac, minimum sensitivity of the big Triac, dU/dt stuff, leakage over (old dirty) PCB's and probably more.