What's the right type of lumber to construct a small TV platform?
I'm not sure of the size of the stand, but you want to make sure your platform is bigger than the stand by enough on each side so that it is stable. 9" wide seems a bit small, given the size of the two TV stands I have, but yours may be different. You may consider securing the platform to your AV center, as well as secure the TV stand to the platform, to reduce the risk of the TV tipping over.

Framing lumber

As for the wood to use, framing lumber and studs are thicker, yes, but the quality is generally lower - they're typically covered up by drywall or tile, after all. They are also almost always pine, a softwood. Since you're going to paint it, looks may not make a difference, but that lumber also tends to have more knots and it can warp more. Additionally, and perhaps most crucially, 2x lumber comes with rounded edges. You'd probably want to square up the edges before gluing to eliminate that rounded edge.

Appearance boards

"Appearance boards" are 1x wood, typically some sort of hardwood (poplar, oak, or aspen), although they do sell pine as well. They're obviously thinner than the 2x wood (half as thick), but will stand up to use better and will generally be flatter and less warped. The edges are also square, so there's no annoying rounded corner to deal with. With some screws, you'll likely be able to get a decently flat glue up, especially if you pick out flat wood to begin with.

With both of those options, though, you're looking at buying 10" wide wood (actual 9 1/4"), which can be quite expensive. Alternatively, you can glue two thinner boards together, but that would require jointing the two edges. If you've got a jointer, router, or table saw, you can do it, but I'm guessing you don't.

Plywood

A third option, which you may have overlooked, is to use (furniture grade) plywood. It comes in 3/4" thickness (other thicknesses available as well), so you would need 5 or 6 layers to get your 4". However, it comes in 4'x8' sheets (both the blue and orange stores carry them for about $45 per sheet; other sizes available, too, as I don't think you'd need a whole 4x8 sheet), so the width is not a problem. Plywood is extremely flat, so the layering is not a problem. The edges have a banded look, but when painted they may not look much different than the rest.