In this case, it's actually referring to both at the same time. Episcopal is the denomination. st.luke is that specific church building which is of the Episcopal denomition.

"St. Luke's Episcopal Church" would be the name of one specific building, and the religious organization that uses it--the congregation, minister, etc. This passage sounds like typical small-town America. The "Episcopal Church" is the name of a Protestant denomination. So is the "Presbyterian Church." Both of them refer to large organizations with millions of adherents in thousands of congregations. In U.S. culture the traditional classification is that "Christians" are divided into "Protestants" and "Catholics." Protestants are divided into "denominations." The Episcopal Church is basically the Church of England, but it broke from the Church of England and changed its name because Church of England clergy had to swear an oath to serve the King of England. The Presbyterian Church has origins in Scotland. Typically, in the United States, most Protestants regard all of the Protestant denominations as being valid. People may choose the church they attend based on social factors, such as how they like the minister or the congregation. The doctrinal and theological differences are considered mostly technicalities. People are almost unaware of them. As children they might learn them in Sunday school, then forget them. This is particularly true of so-called "mainline Protestant denominations:" a vague category, including Baptists, Congregational, Episcopalians, Lutherans, Methodists, Presbyterians, UCC (United Church of Christ), Unitarian. (President Obama is UCC). In the 1950s these were, collectively, the dominant "religion" of the United States. (They've been losing ground to "Evangelicals.") Protestants and Catholics, although they are both Christians, traditionally have regarded each other with some degree of suspicion and distrust. Nevertheless, this audience, which was probably overwhelmingly Protestant, was eager to hear Senator McCarthy, who was (as a U.S. native would guess from his Irish surname) Catholic.

Episcopal is the denomination. 'St. Luke’s Episcopal Church' could refer to either the building or the congregation - in the vast majority of cases they are exactly the same, and it would be unusual to distinguish between them, except when (say) a couple of different congregations meet at the same physical church, or a church meets at a couple of different locations.