Who is obligated to affix a mezuza, the rental or the owner? TB Avodah Zarah 21

The Torah commands the affixing a mezuza on the doorposts of houses and gates as it is written “וּכְתַבְתָּ֛ם עַל־מְזֻז֥וֹת בֵּיתֶ֖ךָ וּבִשְׁעָרֶֽיךָ: -inscribe them on the doorposts of your house and on your gates.” (Dt. 6:9) Of course you recognize this verse because it is the last sentence of the Shema.

According to today’s daf TB Avodah Zarah 21, the obligation to affix a mezuza falls upon the renter and not the building itself. “Rav Mesharshiyya says in response: Affixing a mezuza is the obligation of the resident, rather than an obligation that applies to the house. Therefore, if no Jew lives in a house, it is not subject to the mitzva of mezuza. This means that by renting a house to a gentile one is not removing the right to perform the mitzva from the house.” (Sefaria.org translation)

Rashi ד"ה מְזוּזָה חוֹבַת הַדָּר הוּא explains the basis of this ruling is the understanding of the word בֵּיתֶ֖ךָ - your house. We need to understand that the word house, beitekha to be interpreted as biatekha, your entry. In other words, the place you come in and go out of. The renter who is now living in the house is obligated to affix the mezuza. If the renter is not Jewish, the Jewish owner is under no obligation to affix the mezuza.

Interesting, the Gemara on daf TB Menakhot 34a base a different ruling on word בֵּיתֶ֖ךָ - your house. “The Gemara asks: What is the source for this requirement that the mezuza be affixed to the right side? As it is taught in a baraita: When the verse states: “And you shall write them upon the doorposts of your house [beitekha]” (Deuteronomy 6:9), the word beitekha is interpreted as biatekha, your entry, i.e., the mezuza must be affixed to the side by which you enter, which is from the right.” (Sefaria.org translation)