Due to whom is the modern (i.e. via its universal property) definition of the tensor product, and in which article was it communicated?
My recollection agrees with Mariano. Here's an except from Weibel's History of homological algebra
and from p. 172 of Mac Lane's Homology
and from Whitney's Collected Papers v. 1
is usually credited as the original source.
Very soon after that, Bourbaki crystallized the modern definition.