Choosing TRIAC snubber resistor for multi-purpose switching
TRIACs switch off at (near) zero current.  It is common for switches that switch passively at zero current to experience a voltage step that will cause circuit parasitic inductance and capacitance to ring.  There are 2 problems:  


The peak voltage of the ringing can exceed the rating of the TRIAC.  
TRIACs also have a maximum rated \$\frac {\text {dV}} {\text {dt}}\$ that if exceed will cause the TRIAC to spontaneously trigger.


A snubber, like in your Figure 13 would be used to dampen energy in the parasitic elements.  The inductance will be in the load (\$L_L\$), since it is common to use TRIACs for motor control which are inductive.  The parasitic capacitance is the capacitance of the TRIAC \$C_T\$.  The snubber works by providing an impedance match to the \$L_L\$ \$C_T\$ resonance.  Snubber resistance \$R_s\$ is added to the load resistance \$R_L\$ to match characteristic impedance Zo = \$\sqrt{\frac{L_L}{C_T}}\$.  They tell you to use a higher value for \$R_s\$ for loads with higher inductance because Zo increases with increasing \$L_L\$.  

Typically you will want to use a value for \$C_s\$ that is 10 times \$C_T\$.  For a medium sized TRIAC (one that handles about 10A) \$C_T\$ is often about 100pF.  I didn't see a spec for \$C_T\$ in the datasheet for the NXP BT138 TRIAC.  The best value for the \$R_s\$ is Zo-\$R_L\$.  

Here is a link to an app-note that provides more detail.