To quote from my book, The Bayesian Choice (2007, Section 2.3, pp. 63-64)
The Bayesian approach to decision theory integrates on the parameter
space $\Theta$, since $\theta$ is unknown, instead of integrating on
the sampling space ${\cal X}$, as $x$ is observed. It relies on the
posterior expected loss \begin{eqnarray*} \rho(\pi,d|x) & = &
\mathbb{E}^\pi[L(\theta,d)|x] \\ \tag{1}
& = & \int_{\Theta} \mathrm{L}(\theta,d) \pi(\theta|x)\, \text{d}\theta, \end{eqnarray*} which averages the error (i.e., the
loss) according to the posterior distribution of the parameter
$\theta$, conditional on the observed value $x$. Given $x$, the
average error resulting from decision $d$ is actually $\rho(\pi,d|x)$.
The posterior expected loss is thus a function of $x$ but this
dependence is not an issue, as opposed to the frequentist dependence
of the risk on the parameter because $x$, contrary to $\theta$, is known.
Given a prior distribution $\pi$, it is also possible to define the
integrated risk, which is the frequentist risk averaged over the
values of $\theta$ according to their prior distribution
\begin{eqnarray*} r(\pi,\delta) & = & \mathbb{E}^\pi[R(\theta,\delta)]
\\ \tag{2}
& = & \int_{\Theta} \int_{\cal X} \mathrm{L}(\theta,\delta(x))\,
f(x|\theta) \,\text{d}x\ \pi(\theta)\, \text{d}\theta. \end{eqnarray*} One particular appeal of this second
concept is that it associates a real number with every estimator, not
a function of $\theta$. It therefore induces a total ordering on the
set of estimator s, i.e., allows for the direct comparison of
estimators. This implies that, while taking into account the prior
information through the prior distribution, the Bayesian approach is
sufficiently reductive (in a positive sense) to reach an
effective decision. Moreover, the above two notions are equivalent in
that they lead to the same decision.
A bit further, I use the following definition for the Bayes risk:
A Bayes estimator associated with a prior distribution $\pi$ and a loss
function $\mathrm{L}$ is any estimator $\delta^\pi$ which minimizes
$r(\pi,\delta)$. For every $x\in\cal{X}$, it is given by $\delta^\pi(x)$,
argument of $$\min_d \rho(\pi,d|x)$$ The value $$r(\pi) =
r(\pi,\delta^\pi)\tag{3}$$ is then called the Bayes risk.
Interesting question, I will try to given an answer focused mainly on the history and terminology point of view.
First off, that paper by Berner et al. you mention is far from begin the "first and only ML reference" defining the Bayes classifier. In fact, in that very same paper, the authors cite the book Learning Theory : An Approximation Theory Viewpoint (2007) by Cucker and Zhou as a reference which defines the Bayes classifier.
In said book, the authors indeed define in Proposition 9.3 the Bayes classifier (or Bayes rule) for the binary classification case ($\mathcal Y := \{-1;1\} $) as
$$f^*(x) := \begin{cases}1 &\text{if }\ \mathbb P(Y=1\mid X=x)\ge\mathbb P(Y=-1\mid X=x)\\
-1 &\text{if }\ \mathbb P(Y=-1\mid X=x)>\mathbb P(Y=1\mid X=x)\end{cases} $$
(Which is simply the binary version of the definition you gave) and mention that they give it that name because it is a minimizer of the risk $\mathcal R$, hence for these authors the answer to your question is $ii)$.
Going further back, the earliest reference (I could find) which introduces the notions of Bayes risk and Bayes classifier is Introduction to Statistical Pattern Recognition (1972) by Keinosuke Fukunaga. In Section 3.1 of that book, Fukunaga introduces the problem of binary classification in the framework of Bayesian hypothesis testing :
there are two classes $\omega_1$ and $\omega_2$, and we assume that we know the a priori densities of each class $p_i(X) = \mathbb P( \omega_i\mid X) $ as well as their a priori (unconditional) probabilities $P_i =\mathbb P(\omega_i)$. We can then assign a class to an observation $X$ by using Bayes theorem and computing the posterior densities $q_i(X)$
$$q_i(X) = \frac{P_i p_i(X)}{p_1(X) + p_2(X)} $$
And assigning to $X$ the index $i$ which maximizes that quantity. The author calls this procedure "performing a Bayes test for minimum error", where the error (called the Bayes error), is the expectation of the conditional error $r(X)$ defined as
$$r(X) = \min\{q_1(X);q_2(X)\} $$
Here, the quantity $r(X)$ represents the probability to assign the wrong class to $X$. Quick computations show that the Bayes error (what you would call the risk) is then given by
$$\mathcal R \equiv \mathbb E[r(X)] = P_1\int_\Omega p_1(x)\mathbf 1_{\omega_2}(x)d\mathbb P(x) +P_2\int_\Omega p_2(x)\mathbf 1_{\omega_1}(x)d\mathbb P(x) $$
Two things appear from the above formulation : the Bayes error is by construction the expected 0-1 loss of the Bayes test $\arg\max_{i} q_i(X)$, and that classification rule is optimal by construction. This seems to say that "Bayes decision rule" is the name given to the classifier that minimizes the risk, and also that it is the name of the classifier which assigns to $x$ the most likely class (so both $i)$ and $ii)$ are true).
But in fact, a few pages later, Fukunaga introduces the Bayes decision rule for minimum cost, which now minimizes the expected conditional cost of assigning $X$ to $\omega_i$ given that $X\in\omega_j$ and $j\ne i$ :
$$r_i(X) = c_{i,1}q_1(X) + c_{i,2}q_2(X) \rightarrow r(X) = \min\{r_1(X);r_2(X)\} $$
where $c_{i,j}$ is the cost associated with wrongfully assigning $X$ to $\omega_i$ instead of $\omega_j$.
In that setting, it is true as you noticed that the optimal classifier needs not be the one that assigns to $x$ its most likely class. The author still refers to the optimal classifier as the "Bayes decision rule", so we can conclude that, at least historically, "Bayes classifier" is the name given to the classifier that minimizes the risk, and the name comes from the fact that that classifier is derived from considering the problem from a Bayesian hypothesis testing perspective.
Conclusion : $f^*$ is called Bayes classifier because it is the minimizer of the risk (which is historically referred to as the "Bayes error" in many classical texts such as Devroye et al.'s A probabilistic theory of pattern recognition (1996) or Hastie and Tibshirani's The elements of statistical learning (2009)), but in many cases, it can still be true that $f^*$ is the function that assigns to $x$ its most likely class, so although $ii)$ is always true, scenario $i)$ and scenario $ii)$ may still coincide.
Best Answer
If we let $$ Y := \Theta \\ h := \hat\theta $$ then $R_{SL}(h)$ becomes $$ R_{SL}(\hat\theta) = \mathbb{E}_{p(x,\theta)}\left[L\left(\hat\theta(X),\Theta\right) \right] $$ Using the law of total expectation, \begin{align} R_{SL}(\hat\theta) &= \mathbb{E}_{p(x,\theta)}\left[L\left(\hat\theta(X),\Theta\right) \right]\\ &= \mathbb{E}_{p(x)}\left[\mathbb{E}_{p(\theta\mid x)}\left[L\left(\hat\theta(X),\Theta\right) \right] \right] \end{align} Without loss of generality, if we let $$ X := (X_1,X_2,\dots,X_N) $$ then \begin{align} R_{SL}(\hat\theta) &= \mathbb{E}_{p(x)}\left[\mathbb{E}_{p(\theta\mid x)}\left[L\left(\hat\theta(X),\Theta\right) \right] \right] \\ &= \mathbb{E}_{p(\mathcal{D})}\left[\mathbb{E}_{p(\theta\mid \mathcal{D})}\left[L\left(\hat\theta(\mathcal{D}),\Theta\right) \right] \right] \end{align} Note that $$ \mathbb{E}_{p(\theta\mid \mathcal{D})}\left[L\left(\hat\theta(\mathcal{D}),\Theta\right) \right] = R_B(\hat\theta,\mathcal{D}) $$ and so $$ R_{SL}(\hat\theta) = \mathbb{E}_{p(\mathcal{D})}\left[R_B(\hat\theta,\mathcal{D})\right] $$ This means that $R_{SL}(\hat\theta)$ is just the Bayesian risk averaged over all possible datasets $\mathcal{D}$.