Obviously Greek did not use quotation marks—so what hints are there for determining which parts of the text were quotations of the Corinthian church and which parts are part of Paul's response?
The answer to this question is twofold.
Holman Bible Dictionary has an article that explains how quotations are signified in the New Testament. The most common way to identify quotations in the new testament is by wording (especially verbage) that indicates something has been said or written elsewhere or earlier.
Sometimes also, quotes may be a composition of multiple Old Testament references, and occasionally no verbal hint is given, the text simply repeats something that was said in the Old Testament, or gives indirect quotes and allusions to something from Scriptures. Often, if you compare the Septuagint (LXX or Greek Old Testament) it becomes more obvious something might be a quotation, because the wording and words used will be more similar than comparing Greek to Hebrew. But there's no punctuation used to indicate quotations; as you stated in the question, it's derived from context.
In the particular case of 1 Corinthians 6:13, Paul appears to be referring to an adage or saying from popular culture in Corinth which was used to justify fornication or adultery. It may or may not be an actual quotation of the adage (versus just a reference to it) and many other popular translations do not word it in such a way that they use quotation marks in this verse. The quotation marks appear in some translations to add clarity to the meaning.
As far as why the punctuation was changed on this verse, the Wisconsin Evangelical Lutheran Synod's translation review committee answers this question clearly in their report: