All the water in the house is from the same source - this is usually correct assumption, the only time this might not be correct is if you have some sort of filtration on the water for drinking. If you don't have a filter, then all the water in the house is the same, and as long as you get it from a pipe, it is all drink able. (ie while the water to my faucet and toilet are the same, I will only drink from the non standing source).
If the water is filter, is it is probably whole house. If you only drink bottled water in the house, then you are already proably brewing with bottled water. If the water is good for washing dishes, it is good for washing brewing equipment and good for chilling wort.
In short, if anywhere in the pipes is drinkable, everywhere is. Your pictured splitter looks good. I always like finding valves to stop the flow incase of leaks. I've debated putting a sink in by my washer/dryer and that is exactly how I would do it (with splitters and such). My only concern with the silicon hose and barb is failure over time. So I recommend that you turn it off at your splitter when not in use. When in use, you will be near by in cases of failure and can shut it off before to much water. I think standard water preasure is atleast 1 gallon per minute. Imagine being out for several hours while that leak goes!
My wort chiller I got from a brewsupplier has a barb attachement to the silicon with a hose clamp over it