The equivalence of definitions Riemann integral

This is explained very nicely in Apostol's Mathematical Analysis. In this book Apostol uses Riemann sums to define the definite integral, but he does not use the limit based on $||P|| \to 0$. Rather he uses the concept of finer partition.

Thus if $P, P'$ are partitions of $[a, b]$ then $P'$ is said to be finer than $P$ if $P \subseteq P'$. Adding more points to an existing partition makes it finer. The other concept is called norm of a partition which is defined as the length of largest sub-interval made by the partition. When you add points to a partition to make it finer, then the norm can only decrease (or remain unchanged). Thus finer partitions corresponds to partitions with smaller norms, but the converse does not hold.

It appears from Apostol's presentation that dealing with a limit where norm of partition tends to $0$ is difficult compared to dealing with limit when partitions become finer and finer. And he uses the following definition of Riemann integral:

Let $f$ be bounded on $[a, b]$ and let $P = \{x_{0}, x_{1}, x_{2}, \ldots, x_{n}\}$ be a partition of $[a, b]$. A sum of the form $$S(P, f) = \sum_{i = 1}^{n}f(\xi_{i})(x_{i} - x_{i - 1})$$ where $\xi_{i}$ is any point in the interval $[x_{i - 1}, x_{i}]$ is called a Riemann sum of $f$ over partition $P$. A number $I$ said to be the Riemann integral of $f$ over $[a, b]$ and we write $$\int_{a}^{b}f(x)\,dx = I$$ if for every $\epsilon > 0$ there is a partition $P_{\epsilon}$ of $[a, b]$ such that $$|S(P, f) - I| < \epsilon$$ for partitions $P$ of $[a, b]$ which are finer than $P_{\epsilon}$.

It is now easy to show the equivalence of this definition of Riemann integral with the definition based on Darboux sums (the first definition of your question). This is because the Riemann sums are always sandwiched between the upper and lower Darboux sums and as partitions get finer and finer the lower sums increase and upper sums decrease. If these Darboux sums have same limit (i.e. supremum of lower sums is equal to infimum of upper sums) then obviously the Riemann sums also tend to same limit as partitions become finer and finer. On the other hand if $M_{i}, m_{i}$ are the supremum and infimum of $f$ on a sub-interval $[x_{i - 1}, x_{i}]$ generated by partition then it is easy to choose points $\xi_{i}, \xi'_{i}$ in this sub-interval such that $f(\xi_{i})$ is near $M_{i}$ and $f(\xi'_{i})$ is near $m_{i}$. Due to this we can a find a Riemann sum close to $U(P, f)$ and another Riemann sum close to $L(P, f)$. So if Riemann sums tends to a limit when partitions get finer and finer then the upper and lower Darboux sums also tend to the same limit.

Next Apostol shows the equivalence of his definition (mentioned above) with the definition based on limit of type $||P|| \to 0$ in his exercise. This proof is also available as an answer here.

Also note that in general (i.e. in the context of Riemann-Stieltjes integral) these two definitions are not equivalent and the one which uses the concept of finer partitions is more inclusive than the one based on norm. See this answer for more details.

That the first condition implies the first is immediate, since (using your notation) you always have $m_i \le f(\xi_i) \le M_i$, so the sums in the second definition are caught between the $L$ und $U$ sums.

Edit in response to a comment an additional explanation is necessary here. For this direction it suffices to show that $I^* = lim_{||P||\rightarrow 0} L(f,P)$ and $I_* = lim_{||P||\rightarrow 0} U(f,P)$ Since both parts are similar it suffices to show, e.g., the first equality.

First it is easy to see that for partitions $P\subset P^\prime$ we have $L(f,P)\le L(f,P^\prime)$. A remaining hurdle is that for two partitions we do not necessarily know that one is a subset of the other one. This is resolved by looking at common refinements:

Assume $P$ satisfies $L(f,P) > I^* - \varepsilon$ and $Q$ is an arbitrary partition. We need to show that then there is a refinement $Q^\prime$ of $Q$ such that $L(f,Q^\prime)\ge L(f,P)$ (and, consequently, $L(f,Q^\prime)>I^*-\varepsilon$).

For $Q^\prime$ one can choose the common refinement $R$: if $P=\{x_1,\ldots x_n \}$ and $Q=\{y_1,\ldots y_m \}$ then we just let $R = P\cup Q$. Since this is a refinement of both $P$ and $Q$ we have both $L(f,R)\ge L(f,P)$ as well as $L(f,R)\ge L(f,Q)$

Second edit: the original version was not correct:

For the other direction it suffices to show that if the function is integrable in the sense of the second definition then both $I_*$ and $I^*$ agree with the of the sums from the second definition. Since the reasoning is the same in both cases I'll just look at $I_*$.

So fix $\varepsilon >0$ and a given partition $P$ such that $$|L - \sum_{i=1}^n f(\xi_i)\Delta x_i |< \varepsilon$$ if only the partition is fine enough.

Choose such a partition $P=\{x_0,\dots x_n\}$ and to $[x_{i-1},x_{i}]$ choose $\eta_i\in[x_{i-i},x_{i}]$ such that for $m_i:=\inf \{ f(x):x\in [x_{i-1},x_i]\} $ we have $$0\le f(\eta_i)-m_i\le \frac{\varepsilon}{2n}$$

Then \begin{eqnarray} | L -\sum_{i=1}^n m_i \Delta x_i| & = & |L- \sum_{i=1}^n f(\eta_i)\Delta x_i + \sum_{i=1}^n f(\eta_i)\Delta x_i -\sum_{i=1}^n m_i\Delta x_i| \\ &\le & |L- \sum_{i=1}^n f(\eta_i)\Delta x_i| + \sum_{i=1}^n | f(\eta_i) - m_i|\Delta x_i \\ & < & \frac{\varepsilon}{2} + \sum_{i=1}^n \frac{\varepsilon}{2n}=\varepsilon \end{eqnarray}

If you 'see' that $0 <L -I_*< L -\sum_{i}m_i \Delta x_i$ then you are done here, otherwise it follows easily from the last estimate that the $\sum_i m_i \Delta x_i$ are, for any partition which is fine enough, $\varepsilon $ close to the fixed real number $L$, which of course implies that the $\sup$ over these sums exists and equals $L$ (here you need to use again the fact that you will approach the $\sup$, if it exists, if the width of the partitions goes to $0$).