PE includes buy-out (leveraged and not) and VC transactions. PE is typically any medium to long term equity investment not traded publicly on an exchange. Even this is cloudy now that the PE firms themselves are going public.

LBOs are also not a black and white classification, at least not the way they were in the Gordon Gecko 80's, with varying levels of target-borne debt financing specific to the deal. So while I agree "VCs don't do LBOs", PE does both LBOs and VC deals, with the PE firms doing their own style of fund and deals.

I found this book (though dated) to be a more academic analysis of PE: https://www.wiley.com/en-us/Private+Equity%3A+History%2C+Gov...

don't buy it; try your local library.