Let $X$ be a Noetherian scheme. Is the obvious functor $D(\operatorname{Coh}(X))\to D(\operatorname{QCoh}(X))$ fully faithful?

If this is true then $D(\operatorname{Coh}(X))$ is equivalent to the full subcategory of $D(\operatorname{QCoh}(X))$ consisting of those complexes whose cohomology sheaves are coherent. The bounded above version of the latter statement is here: https://stacks.math.columbia.edu/tag/0FDA

Are any counterexamples known? If yes, how does one "usually" define the unbounded derived category of coherent sheaves?

P.S. It appears that the "modification" of $D(\operatorname{Coh}(X))$ consisting of those quasi-coherent complexes whose cohomology sheaves are coherent is more "useful". This is the version that is relevant for Grothendieck duality; isn't it?