I really do not see the need to use flood coolant on a lathe unless you are doing production work or exotic metals like titanium, and the latter cutting oil may catch fire. As others have mentioned many of the dark cutting oils will quickly discolor exposed metal on the lathe, and if you do not have an overhead splash field will cover your garage door, ceiling and you with coolant. I have made hundreds of systems for different people and I can't recall one person that opted for flood coolant. Just because you have a coolant pump, doesn't mean you need to use it. Even the people that do machining for income, on a manual machine I have not seen flood coolant used routinely. I would get some time on your lathe as is, use a brush or oil bottle, and if you want to take it to the next level use an air/liquid system like the Fogbuster. I really do not see the reason to spend money on 5 gallons of cutting fluid just to try it and then rarely use it after.
If you insist on flood coolant I previously mentioned in your other thread that I have used the KoolRite emulsion oils in my metal bandsaw, it was the only one that lasted more than a few months and didn't turn nasty and stink up the garage. I was in Tucson, and I changed it after a year more because of cleaning the cuttings as opposed to the fluid going bad. Since it is a dilution with water, at around 8-9% a gallon can last a long time, but you must periodically check the specific gravity and add water or coolant. It worked very well in the bandsaw and helped remove the chips, but I now run it dry because I use it less frequently and got tired of the mess and this is with splash shields on the saw. You can also check with David Best, he uses a FogBuster system with a water base coolant that has also lasted. I previously tried Kool Mist and it went bad quickly. I use thread cutting oil on the lathe for turning, but you must wipe it down when done otherwise it will stain. Also Anchor Lube and host of other cutting oils. Save you money and get some time on your lathe before jumping down this pit.