Is the remembrance of Allah, referred to in the Prophet's saying, "And a man who remembered Allah in solitude, and his eyes overflowed," meant to be the recitation of the Qur'an, or the recollection of sins and transgressions and regretting them?

The meaning of the Hadith "he remembered Allah and his eyes overflowed" is that remembrance is with the tongue or the heart in solitude, whereupon the eye weeps out of awe or longing for Allah. Reading the Qur'an is included in remembrance. As for remembering sins and weeping over them, that is a fruit of remembrance, which prompts repentance and seeking forgiveness.