All of this terminology gets used in different contexts and gets confusing.
Technically the aperture is the size of the image circle so APS-C lenses do have a smaller aperture.
F-stops are relative so APS-C has a smaller aperture at the same F-stop.
This is incorrect. See my post on the previous page. Aperture as discussed in camera terms is the relationship of the size of the opening of the aperture blades to the focal length of the lens. What you are describing is the image circle, which is the diameter of the in-focus image plane created by the lens.
The image circle size does not change when you change the aperture. You don't suddenly start cropping images because you've dialed up to f/22. In fact, vignette is more common at WIDER apertures! What really bakes people's noodle is that ever part of the front element of the lens contributes to every part of the image. So the very center of the lens is putting some light in to the very corner of the photo, and the very edge of the lens is also putting some light in to the very center of the image. Weird, but true. Obviously, the angles of the light to do that can get steep. And that's exactly whey wider aperture shots (say F/1.4) have blurry out of focus areas - all that light from all that area can't get bent to the right place and be in focus at the same time. When you stop down to say f/16, now you are taking the light only from the center section of the front element. The light that can bend out to still fill the corners is coming in straighter, and thus more of the image is in focus or close to focus - exactly whey the DOF is deeper! Now you've also stopped collecting light from the outer areas of the front element (the aperture blades are literally, physically blocking it. This is why you have to leave the shutter open longer to get the same exposure.
None of the above changes because there is an APSc sensor behind it instead of a FF sensor. The exposure time, DOF, etc will be the same in the same conditions on both. The only thing that changes is the field of view, since the APSc sensor isn't using the full image circle of a FF lens.
Brian