The nice thing about the logic-level FET is that the Rds(on) is specified at logic levels in the datasheet table. To use a standard FET at logic levels means more calculation.
In general choosing a FET with lower breakdown voltage (BVDSS) will give you lower Rds(on). Unlike Rds(on) BVDSS increases with temperature so do add margin for temperature increase.
The LM317 in the Instructables circuit regulates the current through the LED using closed-loop feedback. IIRC the LM317 maintains a voltage of 1.2V between the Out and Adj pin. The current through the resistor is 1.2V / R. Since no current goes into the adjust pin all of the current has to go through the LED. The LM317 is acting like a variable resistance.
How are you proposing to shutoff the LM317? I do not see a practical way to do it
(* jcl *)