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.
- Quotations From the Scripture/Word: “as the Scripture has said...”, “What says the Scripture? ...”, “As it is written...”
- Quotation from OT prophets: e.g., “For Isaiah says...” (David, Moses, etc)
- Fulfillment of OT prophesy: e.g., “that it might be fulfilled”
- Quoting God or the Holy Spirit: “God has said...”, “The Holy Spirit says...”
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:
In 1 Corinthians 6:13, NIV1984 punctuates in such a way that Paul is the one who teaches that God will destroy both food and the stomach. That understanding gives rise to unwarranted speculation about the nature of the resurrected body. That in turn creates tension with Luke 24, where we see that the body of the risen Jesus is capable of eating food (even if it is not dependent on food). NIV2011 pays closer attention to the way Paul provides answers to Corinthian thinking on a nearly clause-by-clause basis. Accordingly, NIV2011 punctuates in such a way as to make it clear that it is the Corinthians who say God will destroy both food and stomach: "You say, 'Food for the stomach and the stomach for food, and God will destroy them both.' The body, however, is not meant for sexual immorality but for the Lord, and the Lord for the body."