I do something similar to this. I use 15 gallons of water, all of which gets captured and reused, plus a 5lb bag of ice and 2 frozen 2l bottles, to cool my average 5 gal brew batch, using a similar cheap submersible fountain pump.
I fill three 5 gallon buckets with "cold" water from the tap (74 degrees last I checked), and pour them one at a time into another bucket which has the pump. The outflow is captured into other buckets on the other end for reuse. Nothing goes down the drain. Two buckets (10 gal) of my "barely cold" tap water get the wort down to nearly 100 degrees. Then the third bucket goes in and the outflow goes into the pump bucket to recirculate. Adding ice sooner means the relatively high temp of the tap water kills your ice far too quickly to do enough good. At the high wort temps, the temperature differential from plain tap water does the job just fine. You wouldn't see a huge difference in cooling time by using colder water, but once your ice is melted, cooling the wort the rest of the way with warmer water is a very slow process due to the small temp differential.
Using my method with a generic 25' copper coil immersion chiller, I consistently get my wort down from boiling temp to 85 degrees in a half hour. Transferring the wort from the boil pot to the fermentation vessel (and accompanying aeration) drops it another 5 degrees to 80, which is my normal pitching temp.
Here are some notes from my last brew session, starting when the boil ends:
0:00 turn off heat and begin pumping bucket #1 water though the cooler.
0:05 wort temp is 165
0:10 wort temp is 135
0:15 wort temp is 110
Add water as needed. You'll be working on the 2nd bucket by now. (I didn't log the exact transition point)
At this point I make sure the ice is staged and ready. When the wort temp hits 100, I put the ice in the pump bucket, and move the outflow hose to the pump bucket as well, to recirculate the water. No addition water is used for cooling.
0:20 wort temp is 100, add ice.
0:25 wort temp is 95
0:30 wort temp is 85
Begin Transfer to fermentation chamber.
After the yeast is pitched, I have three 5 gal buckets of water at varying temps (from quite hot, to tepid) on hand, for cleanup, watering plants, laundry, whatever.
