To what are these three words referring?

Maseches -

Rashi: Not to admit that he knows a certain Maseches

Puria -

Rashi: Respond in the negative if someone asks if he had relations last night;

Tosfos: Puria means that people do not usually ask in the way Rashi explained it. Rather, if they ask him if he slept on a particular bed, he may change his words, lest they find Keri (semen) on the sheets, which would disgrace him.

Maharsha: Puria means Purim, meaning that a Rabbi may pretend to be more drunk on Purim than he really is.

Ushpiza -

Rashi: Not to praise his host in front of people who are not upstanding.

Maharal: A guest should not discuss the private behavior and matters of the host with others, as a person is not interested in others knowing their private matters.

Why did Chazal specifically choose these three examples?

Shita Mekubetzes: They represent three different traits that are fitting for a Talmid Chacham to have - humility (Maseches), modest behavior (Puria) and appreciation and a desire to help others (Ushpiza).

Iyun Yaakov: Chazal describe that the Rabbanan would leave them homes and travel to learn Torah. Our Gemara refers to when those Rabbanan would return to their homes and would be questioned as to the conditions that they endure. In order not to discourage others from travelling to learn, they may change their words. When asked about whether they studied a difficult Maseches, they may say that it was not so difficult. When asked whether they slept on a bed or on the ground, they may say that it was on a bed (even though Chazal say that the way of Torah is to sleep on the ground, and this is what they did). When asked whether they were hosted well, given good food and drink, they may say that it was good (even though Chazal describe their food as eating bread dipped in salt and drinking water by measure, and this is what they endured).

Why did the Gemara use the phrase 'change their words' rather than 'lie'?

Anaf Yosef: They would never lie. On occasion, they change their words to a phrase that could be taken two ways. The speaker is thinking about the truth and if the listener mistakenly thinks of the other meaning, so be it.

Concerning changing his words about a Maseches, does the Gemara not say that Divrei Torah should be 'sharp in your mouth', so that if asked a question, one can answer without hesitation?

Tosfos R' Peretz, Ritva, Toras Chaim: That Gemara is referring to a Torah authority who is asked to make a ruling or answer a specific question in learning. The permission to 'change his words' applies when the questioner is asking someone else or asking without a need to receive an answer.

Does the Gemara not say elsewhere (Yevamos 65b) that one may also change one's words 'Mipnei haShalom' - for the sake of peace? Why is that not listed here?

Shita Mekubetzes-Shita: That does not need to be listed as it is a Mitzvah.

Tosfos: These three are also examples of 'Darchei Shalom'.

Shvus Yaakov: These three only apply to a Talmid Chacham, but every person should change their words for the sake of peace.