You're describing the same thing from different perspectives.

Bond prices are down, this means yields grow, and thus become more attractive investments than high P/E stocks.

I think the reason this sounds unintuitive is because the bond market isn't exactly a "free market" in the general sense. The Fed is manipulating the money market to set bond yields, so from a causative point of view, the yields rise first, which cause a drop in bond prices, which make buying bonds attractive for investors, who then, as a result, shift assets from stocks to bonds.

(disclaimer: just speaking from knowledge gleaned from, among all things, youtube videos :D definitely not an economist !)