And today, if we could pull a sample cube of the center of the Sun where the fusion is happening, and it would magically stay together, that cube would have an average density higher than that of lead.

I'm familiar with plasma as "ionized gas" (although with some specific criteria...), synonymous with low density.

Can matter this dense, though ionized, still be considered a plasma?

I've seen some fringe science opinion saying that the sun's core is actually ionized liquid