Is this resistor 1/8th watt or 1/4 watt?

You can increase the power rating of any resistor (double or more) by leaving additional lead-in wire (I occasionally loop the leads) raising the resistor off the PCB and/or drilling a hole or removing the PCB substrate from underneath the resistor.

If there is a critical temperature rating for a given component one would expect the circuit designer would specify under what conditions the rating is made and exactly how the component needs to be installed. In many instances the rating of a resistor is to suit the PCB hole spacing (power rating is relative to size), and the power rating itself isn't important because the dissipation is most like way below specified maximum.