You can define the Riemann integral in different ways. The one you are referring to, with the infimum and the supremum, uses the so-called Darboux sums. I will write two equivalent definitions of the Riemann integral and the usual definition of the Riemann-Stieltjes integral. Hopefully, with the second definition of the Riemann integral in mind, it will become clear that Riemann integrable implies Riemann-Stieltjes integrable.
Definition 1 of Riemann integral: Let $f:[a,b] \to \mathbb{R}$ be a bounded function. Consider a partition $P = \{a = x_0 < x_1 < \dots < x_n = b\}$ and define the following sums:
$$L(f,P) := \sum_{i=1}^n m_i (x_i - x_{i-1}) \quad \text{and} \quad U(f,P) := \sum_{i=1}^n M_i (x_i - x_{i-1}),$$
where $m_i = \inf_{t \in [x_{i-1},x_i]} f(t)$ and $M_i = \sup_{t \in [x_{i-1},x_i]} f(t)$. Then, let $\mathcal{P}$ be the set of partitions of $[a,b]$ and set
$$\mathcal{L} := \sup_{P \in \mathcal{P}} L(f,P) \quad \text{and} \quad \mathcal{U} := \inf_{P \in \mathcal{P}} U(f,P).$$
If $\mathcal{L} = \mathcal{U}$, then we say that $f$ is Riemann-integrable over $[a,b]$ and denote its integral by $\int_a^b f dx$.
Definition 2 of Riemann integral: Let $(P_n)$ be a sequence of partitions $\{a=x^n_0 < x^n_1 < \dots < x^n_n = b\}$ such that $|P_n| := \max_{i} |x_i - x_{i-1}|$ goes to zero as $n \to \infty$. The number $|P_n|$ is typically referred to as the mesh of $P_n$. Now, consider the following sums:
$$S(P_n) := \sum_{i=1}^n f(\xi_i) (x_i - x_{i-1}),$$
where $\xi_i \in [x_{i-1},x_i]$ is an arbitrary test point over which $f$ is defined. If the limit $\lim_{n\to \infty} S(P_n)$ exists, then we say that $f$ is Riemann-integrable over $[a,b]$ and denote its integral by $\int_a^b fdx$.
Now, one can show that these two definitions are equivalent. There are two key insights to see this. First, note that we can "sandwich" the sums of Definition 2 with the lower and upper sums of Definition 1. Precisely,
$$L(f,P) := \sum_{i=1}^n m_i (x_i - x_{i-1}) \leq \sum_{i=1}^n f(\xi_i) (x_i - x_{i-1}) \leq \sum_{i=1}^n M_i (x_i - x_{i-1}) =: U(f,P).$$
Second, given a sequence of partitions with vanishing mesh size $(P_n)$, we may construct a new sequence of partitions $(Q_n)$ with vanishing mesh size such that $Q_{n+1} \subset Q_n$, i.e. $Q_{n+1}$ is a refinement of $Q_n$. You can see this post for details or Apostol's Mathematical Analysis.
Regarding the Riemann-Stieltjes integral,...
Definition of Riemann-Stieltjes integral: Consider two real-valued functions $f:[a,b] \to \mathbb{R}$ and $g:[a,b] \to \mathbb{R}$. Additionally, let $(P_n)$ be a sequence of partitions such that $|P_n|\to 0$ as $n \to \infty$. Define the sums
$$S(f,g,P_n) := \sum_{i=1}^n f(\xi_i)\big(g(x_{i}) - g(x_{i-1})\big).$$
If the limit $\lim_{n\to \infty} S(f,g,P_n)$ exists, then we say that the Riemann-Stieltjes integral of $f$ against $g$ over $[a,b]$ exists and we denote it by $\int_a^b f dg$.
Looking at Definition 2 of Riemann integral it should be clear now that the Riemann integral is a particular case of the Riemann-Stieltjes integral. Specifically, the Riemann-Stieltjes integral is a Riemann integral only when $g(x) = x$, i.e. the integrator is the identity function. If a function is Riemann-integrable, then it is immediately Riemann-Stieltjes integrable. The converse doesn't even make sense because with Riemann integrals you do not consider different integrators.
