If you used a premixed coolant from a major producer, like Peak, Preston et al, they use distilled water to mix it. The coolant you drained was still good as far as the anti-freeze properties were concerned. It's the anti-corrosion you are renewing. The dissimilar metals like to create electrolytic corrosion. As for burping it, if you opened the heat control, let it run until you get heat and the thermostat opens. The Mazda system will burp itself. Once the thermostat opens, fill the reservoir to the hot mark and shut the car off. Close the cap and you're done.

Be glad you don't have a BMW. The last one I worked on was a real bear to get the air out of. I got it done, and the owner was happy as the last guy to work on it almost ruined the engine.

 

If you are flushing the system then you need to run clean water through the system until it runs clear.
Antifreeze properties degrade over time so replacing and flushing the system is the only correct way to insure a cose to completely replacing the coolant.

"BURPING" coolant seems to be a confusion with not only DIY but mechanic's.... The easiest way to ensure that your cooling system is to refill with the radiator cap at at least 2-4 degree upward incline. There is ALWAY air in ALL cooling systems another funny misconception ... remember coolant has 2 main ingredients one is water or H2O. H2O means that each watermoleculecontains one oxygen and two hydrogenatoms.
There are some engine that the head designs were not the best and had a tendency to have air pockets that when the oxygen was not removed could super heat the moisture creating hot spots and fluctuation in the coolant temperatures. These were for the most part rare and most auto manufactured either changed their head design for the production of the engine or gave more details on how to fill the cooling system. As far as I am aware of no MAZDA from manufacture 1990 and ner ever had a head problem effecting the cooling system?

Seems you indicated the use of MAZDA coolant you do not technically need to add any water.

When in doubt or question any service method it is always best to use the FACTORY SERVIVE MANUAL or if available Haynes, Chilton, Clymer service manuals and NOT general internet information that are often filled with flaws!