Yes, I use a turkey baster and do what you do. I do it at every oil change (over kill) to keep the fluid fresh. Use the fluid specifications recommended in the owners manual.
Ford Motorcraft Mercon V @ Walmart
Skip syringe and disconnect hose to discharge all of the fluid in the reservoir -most of the fluid is in the rack. OR better, let the power steering pump do the work for you and actually flush it all.
https://www.mazdas247.com/forum/showthread.php?123845663-Power-Steering-Oil-Change
Ford Motorcraft Mercon V @ Walmart
Skip syringe and disconnect hose to discharge all of the fluid in the reservoir -most of the fluid is in the rack. OR better, let the power steering pump do the work for you and actually flush it all.
https://www.mazdas247.com/forum/showthread.php?123845663-Power-Steering-Oil-Change
The keyword is "equivalent" and understanding that power steering fluid serves as hydraulic fluid that lubs the electric pump/seals and not nearly as complex as the transmission (temp changes/lub mechanical parts, modifiers). It's always safe to follow the dealer but IMO M-V/M 5 is a waste as PS fluid.
Mazda Owners Manual
2008 Power steering fluid: ATF M-III or equivalent (e.g. Dexron II)
2012 Power steering fluid: Mazda Genuine ATF M-III , M-V, or equivalent (e.g. Dexron II)
most places charge around $100-$125 for a power steering flush