This is a great question. Let me start with limits and discuss lax limits later. Given a $D$-shaped diagram $X$ of model categories (where $D$ is a small category), one can ask whether the two ways (Barwick vs Bergner) of computing the homotopy limit agree. As Bergner points out on page 7, strictly speaking, the two methods do not agree (Barwick's outputs a right semi-model category while Bergner's outputs a specific subcategory of a model category), but they do agree once you pass to the level of $\infty$-categories. More on this below. Next, you can ask whether the underlying $\infty$-category of the homotopy limit of $X$ agrees with the homotopy limit (in an $\infty$-categorical sense) of the diagram of $\infty$-categories where you first replace every model category in $X$ with its underlying $\infty$-category, then take the homotopy limit of that diagram. The answer, again, is that they do agree.
First, let's fix some terminology, that we need for comparing the different models. In the model of quasicategories, you can compute the homotopy limit of a diagram of quasicategories several equivalent ways: using the Joyal model structure, using weighted limits, using the model structure on marked simplicial sets, or using pseudo-homotopy limit cones and $\infty$-cosmoi. See also these notes of Emily Riehl. In the model of simplicial categories, or the model of complete Segal spaces, one can again compute homotopy limits using the corresponding model categories, or could do so by pushing everything into an equivalent model like quasicategories, or could do so internally without reference to model categories as in Bergner's Theorem 2.5 of the paper linked in the OP.
Now, the paper of Bergner that you cite explicitly carries out a comparison between the homotopy limit of model categories and the homotopy limit of complete Segal spaces (Section 5). They are equivalent. Furthermore, the model category of complete Segal spaces is Quillen equivalent to the model category of simplicial categories (see Theorem 2.4 in Bergner's paper) and that is Quillen equivalent to the Joyal model structure on $sSet$ that encodes quasi-categories (see chapter 1 of HTT). Now, the whole point of a homotopy colimit is that if you replace your diagram by a weakly equivalent diagram, the homotopy colimit is the same, up to weak equivalence. If you had two ordinary categories $C$ and $C'$, and an equivalence of categories $F:C\to C'$, a diagram $J: D\to C$, and then you created the corresponding diagram in $C'$, it would be the case that $F(\lim D) \cong \lim F(D)$. The same thing is true in $\infty$-categories, thanks to the universal property.
What about Barwick's paper? Well, Bergner also points out (on page 7 of her paper) that her method of computing the homotopy limit of a diagram of model categories agrees with his, at the level of $\infty$-categories. Barwick's method involves right Bousfield localization and might output a semi-model structure. Bergner's method involves restricting to a subcategory of the lax limit. But these two ways of getting at the homotopy limit do agree on the level of $\infty$-categories, e.g., because one could fibrantly replace everything before right Bousfield localizing, without changing the underlying $\infty$-category. So, for homotopy limits, all ways of computing them agree.
Ok, so what about lax limits? Again, let $X$ be a $D$-shaped diagram of model categories, with one $M_\alpha$ for every $\alpha \in D$ and left Quillen functors $F_{\alpha, \beta}^\theta: M_\alpha \to M_\beta$ for every $\theta: \alpha \to \beta$ in $D$, plus the obvious compatibility. One could equivalently use right Quillen functors because every left Quillen functor has a right adjoint. The Bergner and Barwick methods of computing the lax limit (as a model category) agree. Bergner's Definition 3.1 on page 5 of her paper denotes the lax limit as $L_D X$. An object is a family of pairs $(x_\alpha, u_{\alpha, \beta}^{\theta})$ where $x_\alpha \in M_{\alpha}$ and $u_{\alpha, \beta}^\theta: F_{\alpha, \beta}^\theta(x_\alpha) \to x_\beta$ is a morphism in $M_\beta$. Morphisms are levelwise. Barwick's lax limit is $Sect^R X$ from his Definition 2.21. An object is a family of pairs $(x_\alpha, \phi_f)$ where again $x_\alpha \in M_\alpha$ and $\phi_f: x_\alpha \to f^* x_\beta$ where $f: \beta \to \alpha$. The only difference is that we're using the adjoint version, prioritizing right Quillen functors (right adjoints to the F_{\alpha, \beta}^\theta) over left Quillen functors. Barwick's remark 2.22 illustrates the equivalence with Bergner's style of lax limits. Bergner's Section 6 shows that the $\infty$-category underlying the lax limit of $X$ is equivalent to the $\infty$-categorial lax limit of the diagram of $\infty$-categories underlying $X$. Also, the lax limit in category theory is preserved under equivalence of categories, and similarly the $\infty$-categorical lax limit is preserved under changing your model for $\infty$-categories (e.g., between quasi-categories and complete Segal spaces).
Now, Remark 4.4 in the cited paper of Horev and Yanovski (HY) points out that their notion of lax limit of a diagram of $\infty$-categories (via coCartesian fibrations) agrees with that of Gepner, Haugseng, and Nikolaus (GHN; the definition is via the twisted arrow category, but the introduction spells out the connection to the Grothendieck construction and coCartesian fibrations). All that remains is to compare the $\infty$-categorical lax limits with model categorical lax limits. Let's use Definition 4.1 of the HY paper: the lax limit of a diagram of $\infty$-categories is defined to be the $\infty$-category of sections of the corresponding coCartesian fibration. This can be thought of as the $\infty$-category underlying the model structure in Barwick's Theorem 2.28 (or, equivalently, underlying Bergner's $L_D X$). So, the two approaches agree! This is what Barwick was getting at in his Remark 2.24 that you mentioned. The comparison is carried out explicitly in the paper Lax limits of model categories by Yonatan Harpaz, using the theory he and Prasma worked out regarding the Grothendieck construction in model categorical contexts.
As you think more about lax limits, I encourage you to carefully study the introduction to the GHN paper, which does a great job summarizing the equivalent ways to think about lax limits. It would also be good to review the literature in classical category theory, cited by GHN. In particular, I encourage you to write out the Grothendieck construction associated with a diagram of ($\infty$-)categories and see why the section approach agrees with the weighted limit approach.