DIAC datasheet's circuit is not in series with TRIAC's Gate
That's not a &quot;diac&quot;, it's a photo triac.
The gate-MT1 resistor is to deal with leakage and bias current in the photo triac and/or zero crossing circuit. It effectively lowers the sensitivity of the triac a bit by conducting some of the current that would otherwise go to the gate.
Many triacs are insensitive enough that it won't make any difference.
Worst case will usually be at high ambient temperature (for both the triac and the opto-isolator). So you want to pick a value that is low enough that the triac cannot trigger when it is supposed to be off, but does not conduct so much current that the circuitry cannot drive it. If you make the resistor too low, then R2/R3 (1st schematic) or R13 (2nd schematic) will burn up because the triac won't trigger at all.