The book A History of Mathematics: An Introduction
by Victor J. Katz says:
"...probably the most famous mathematical technique coming
from China is the technique long known as the Chinese
remainder theorem. This result was so named after a
description of some congruence problems appeared in one
of the first reports in the West on Chinese mathematics,
articles by Alexander Wylie published in 1852 in the
North China Herald, which were soon translated into both
German and French and republished in European journals..."
(page 222)
This seems to suggest that the name "Chinese Remainder Theorem"
was introduced soon after Wylie's 1852 article.
But the book Historical Perspectives on East Asian Science,
Technology, and Medicine, edited by Alan Kam-leung Chan,
Gregory K. Clancey and Hui-Chieh Loy says:
"A. Wylie introduced the solution of Sun Zi's remainder
problem (i.e. "Wu Bu Zhi Shu") and Da-Yan Shu to the
West in 1852, and L. Matthiessen pointed out the identity
of Qin Jiushao's solution with the rule given by C. F. Gauss
in his Disquisitiones Arithmeticae in 1874. Since then it
has been called the Chinese Remainder Theorem in Western
books on the history of mathematics."
This is ambiguous, as it is not clear whether the author
means that the name "Chinese Remainder Theorem" came into
use after 1852 or after 1874. But the phrasing does suggest
that the name came into use before 1929.
In 1881, Matthiessen published the following article:
L. Matthiessen. "Le problème des restes dans l'ouvrage
chinois Swang-King de Sum-Tzi et dans l'ouvrage Ta Sen
Lei Schu de Yihhing." Comptes rendus de l'Académie de
Paris, 92 :291-294, 1881.
But does the name "Chinese Remainder Theorem"
("le théorème chinois des restes") appear in this article?
The second result you're talking about is also sometimes called the Chinese remainder theorem, and can be derived from the Chinese remainder theorem for rings by "tensoring the CRT isomorphism" with $A$. Explicitly, (1) gives
$R/\prod_{k=1}^n I_k\cong\prod_{k=1}^n R/I_k$
via the natural map. This is an isomorphism of rings as well as an isomorphism of $R$-modules. Therefore, upon tensoring with $A$, it becomes
$A/\big(\prod_{k=1}^nI_k\big)A\simeq \prod_{k=1}^n A/I_kA$
via the natural map, using the canonical isomorphism $R/I\otimes_RA\cong A/IA$ as well as the fact that tensor product commutes with finite direct products. It follows that the kernel of $A\rightarrow\prod_{k=1}^n A/I_kA$, which is clearly $\bigcap_{k=1}^n I_kA$, is equal to $\big(\prod_{k=1}^n I_k\big)A$. So, you've derived (2) from (1). Keep in mind that (2) is an isomorphism of $R$-modules, while (1) is an isomorphism of rings (as well as $R$-modules).
Best Answer
Parallel computation: Suppose you have a huge computation to do that involves adding, multiplying and subtracting integers. Possibly also dividing but, if so, only division by numbers in a finite set S which you already know.
Choose primes $p_1$, $p_2$, ..., $p_r$ which do not divide any element of $S$, and such that $p_1 p_2 \cdots p_r$ is surely larger than your answer. Split your computation over $r$ processors, the $i$th of which computes the answer modulo $p_i$. Use CRT to put your answer back together in the end.
This was the method used in the recent computation of the Kazhdan-Lustig-Vogan polynomials of E8.