Who coined the term "baseball diplomacy" (and when)?

My research suggests your source was not just off in claiming the phrase was first used in the 1970s: your source was off by more than half a century. The most generous interpretation of the source quote, wherein the sense of the phrase as describing

specific efforts at using high-profile exhibition baseball games to generate political goodwill between the Washington and Havana governments

is intended by "first used in the 1970s", is contradicted by the much earlier appearance of that sense of the phrase in the Google Books corpus in a play published in 1916:

Here's an account of a baseball game in Cuba, between the Giants and the Almendares. There are nine American words to one of Spanish, and there's not an English word in it. The difference in language is what keeps people apart. ... You'd better abandon warship diplomacy and dollar diplomacy for baseball diplomacy ... baseball, that's the true sport of democracy ... Imagine trying to raise armies during a World's Championship series.

(The Wastrel Hoard: A Drama of the Greater Love, Frank Hendrick, Puritan Play Company, 1916.)

Aside from the appearance in that 1916 play with the specific sense cited by your source, the phrase appears in freely available electronic popular press corpora from the late 1800s with more general senses. The first such instance I found was this:

(New-York Tribune, August 31, 1887.)

Instances of the phrase recur in press corpora searchable with Elephind.com between 1887 and 1931. The last such is an obituary in 1931.

These finds, to be sure, don't take us far toward discovering

1. the specific year the term was coined;
2. who coined it.

My best surmise, for 2, is that the phrase was a 'natural' phrase to be used with reference to machinations surrounding baseball, and so that use by players or managers was adopted by writers for the popular press. For 1, the evidence suggests the phrase was in use for some time, possibly as long as a decade or more, before its appearance in The New York Tribune in 1887.