this is my older machine coolant and spindle cooling I made few years ago, you can see how hoses ... this spindle is not so "nise looking" but it is good (45k rpm) pump is also tiny - exact like yours
("The water is pumped using a small, water feature pump and is more than adequate for the job.") - yes, some of these tiny 12v pumps are very good!
Pump is electronically controlled ...
You can use how many closed loop water-systems you like - I prefer just all in one
cheers,
herbert