That there was a Greek word for Aramaic (Suristi) and Luke chooses to use the word for Hebrew in these places (tae Hebraidi dialectow) implies that he meant Hebrew instead of Aramaic.
Paul, being a Pharisee and trained by Rabban1 Gamaliel the Elder, would certainly know Hebrew. My answer here shows from literature and archeology that Mishnaic Hebrew was used by the common people in the Land during the time of Jesus and Paul. So nothing precludes it from being Hebrew (the speaker and the audience both know Hebrew).
Also, Paul's accusers in chapter 21 are saying he is unorthodox. One way to add to his orthodox credentials (so to speak) would be to use the Holy Language. While that alone wouldn't convince them, it would help his case. The opposite, Paul being unable to speak Hebrew, would drastically hurt his case.
We also see that it was Hebrew by the crowd becoming very quiet. Israelites were raised with a reverence for the Hebrew language which was called "the tongue of Torah," "the tongue of angels," "the tongue of Heaven," and, most importantly, "the holy tongue" (Sifre to Deuteronomy 333; BT Shabbat 115a; Bava Batra 82a; Hagigah 16a, etc.; Bereshit Rabba 18:4; Kohelet Rabba 7:8, etc.).
Paul addresses them as "Men, Brothers, and Fathers." Stephen also uses the same phrase when he addresses a similar crowd in 7:2. "Brothers" shows that Paul sees himself in the same Abrahamic covenant as they are. "Fathers" is a title of honor applied to those in authority (cf. 2 Kings 13:14; Nehemiah 9:32; Daniel 9:6; Matthew 23:9; Acts 22:1). This usage was common in rabbinic circles, as the phrases Pirkei Avot ("Chapters of the Fathers/Religious Authorities," a tractate in the Mishnah), Avot d’Rabbi Natan ("The Fathers/Religious Authorities according to Rabbi Nathan," another early rabbinic work), Av-Bet-Din ("Father/Leader of the House of Judgment [i.e., the Sanhedrin]", a title used for the head of the Pharisees) all attest. Religious instruction was expected to be in Hebrew; his use of this phrase would imply this and the following are in Hebrew.
1The title Rabban instead of Rabbi means that Gamaliel was recognized as the greatest teacher of his generation. He was the first of only seven to be granted it.