There are several definitions of a quasi-equivalence $\newcommand{\T}{\mathscr{T}}F : \T \to \T'$ of DG categories in the literature, e.g.

(i) the induced functor $H^0(F) : H^0(\T) \stackrel{\sim}{\to} H^0(\T')$ is an equivalence (Bondal-Kapranov 1990);

(ii) $H^0(F)$ is an equivalence and $H^*(F)$ is fully faithful, i.e. $$ \newcommand{\Hom}{\mathrm{Hom}}\bigoplus_n H^n(\Hom_\T(X,Y)) \stackrel{\sim}{\longrightarrow} \bigoplus_n H^n(\Hom_{\T'}(F(X),F(Y))) $$ is an isomorphism for all $X,Y \in \T$, $n \in \mathbf{Z}$ (Drinfeld 2002);

(iii) $H^0(F)$ is an equivalence and for all $X,Y \in \T$, $n \in \mathbf{Z}$, $$ H^n(\Hom_\T(X,Y)) \stackrel{\sim}{\longrightarrow} H^n(\Hom_{\T'}(F(X),F(Y))) $$ are isomorphisms (To\"en 2004, Keller 2006).

It is obvious that (iii) implies (ii) and (ii) implies (i), but when are the reverse implications true? Is it obvious when $\T$ and $\T'$ are *(strongly?) pretriangulated*? In particular, I am working with the DG category $\mathbf{L}_{\mathrm{pf}}(X)$ of perfect complexes of sheaves on a smooth proper scheme $X$, and in this case I would hope that all definitions are the same.

I apologize if this question is too elementary for MO.