but my specific queries about this particular T2 tester are:
560 Ohm resistor specification (.125 watt? 3 watt? Which watt?)?
IMHO any 3 or 5mm (diameter) general LED will do.
If you prefer white or blue LEDs the resistor value need to be lowered (to 330 Ohms),
because these LEDs have a higher forward voltage, and the emitted light will be weak.
Despite the description is good, the LED is pictured with wrong polarity on the site, the anode (+) should
face to the resistor, while the cathode should face to terminal #1 of the T2 connector,
so if you connect according the drawing, it won't work (but at least won't do harm either).
Regarding the wattage: driving one LED not requires too much power, a 0.125W resistor will be fine,
it's small enough to easily fit into a shrink tube, pictured on the site.