Eisenhower's administration pushed the notion that being "religious" = being a "patriotic" or "good" American. While it was publicly presented as irrelevant what religion was involved, it was pretty much nudge-wink white Anglo-Saxon protestantism. In those days Catholicism was considered highly suspect, and certainly no one would tolerate a "weird" religion like Islam or Buddhism. Even relative public tolerance for such things was still over a decade away.

This false equivalency was based on the notion of "godless communism" as the main bogeyman and existential threat to "American values". Communism has been so thoroughly conflated for so long with atheism (and, in fact, a particular, imagined "god-hating" form thereof) that to this day, it's still a widely held misconception. Doubtless this obsession with Communism (or really any unfamiliar or impure in terms of capitalism and therefore scary political idea) was a lingering product of McCarthyism.

The Law of Unintented Consequences kicked in with a vengeance from there and fundamentalist Christianity became a sort of litmus test in the minds of its own adherents for being a god-approved capitalist and Good Citizen. Beginning in the mid 1950s and building steadily through at least the end of the 20th century, the evangelicals become more a political than a religious movement -- though always of course with overt religious rationalizations.

Today I think the evangelical star is fading and current ideological polarization on the part of the religious / Republican complex is being partly driven by a realization of this impending loss of hegemony.

So I would say Eisenhower's statement about religion was not about being thorough or diligent apart from dogma. Eisenhower's administration was the source of "in god we trust" on coinage, "under god" in the Pledge of Allegiance, etc. It was very much about being thoroughly religious in the primary, "god-fearing" sense.