Yes, the air gets churned up into fine foam by the action of the pump. While it's foamy the fluid doesn't work well, which is another reason to lift the front of the car. I haven't figured out a better way to work out the air.

Turn the steering back & forth at least 2 or 3 times, then let the fluid settle. Then do it again a couple times. That's to get fresh fluid all the way to each end of the hydraulic cylinder in the steering rack. Each time you let it settle, the air comes to the surface not only in the reservoir, but in the rack & the lines too. It's kind of a tedious pain working all the air out.

When it's cold & you first start up, it's not too unusual for the power steering to be like that. I think the pump needs a few more seconds to wake up. I'm talking about spinning the steering wheel fast (like getting out of a parking spot) immediately when you start.

As the belt gets old, it gets hard & slippery, so more likely to slip. Temptation is to over-tighten the belt, but that's not a good thing to do.

It's hard to figure out blaming the belt vs. the pump itself, but the belt is cheap & easy to replace.