Sump/refugium and closed-loop are two separate things. A sump/refugium is a separate tank (usually sitting under the main tank). Water drains into the sump through an overflow box in the main tank (a 125 will most likely have one built in), passes through the sump and the refugium, and is then pumped back into the main tank with a water pump.
A closed-loop is for water movement and needs to be drilled for separately, and utilizes an external water pump. You will have a hole through which water will drain into the external water pump and will then be pumped back into the tank by the same pump. It's a closed loop because water is looping in and out your tank through the external pump. People have, I think, set up closed loops without drilling their tanks. They just pvc pipes or something hanging on the back to drain water out and pump it back in.
Also, your stock list is a bit too much for that tank. Triggers are extremely agressive fish, especially the Clown Trigger, and will not only outgrow the tank, but will likely kill everyone else in there. The Tang you mention will also outgrow your tank. The only Tangs I'd suggest for a 125 are Yellow, Hippo, Kole, Convict, or Purple.