Is Friday prayer counted for one who enters the mosque after the first Adhan while the Imam is praying the two rak'ahs, or does the recording of attendees stop with the Imam's ascent to the minbar or with the first Adhan, even if the Imam has not ascended the minbar?

One must differentiate between the immense reward for initiating early attendance to Jumu'ah and merely catching it. Missing early arrival does not entirely deprive a person of the reward of Jumu'ah, but rather deprives them of a great reward, as stated in the hadith (definition): "Whoever performs ghusl (definition) on Friday like the ghusl of janabah (definition), then goes early, it is as if he sacrificed a camel. And whoever goes in the second hour, it is as if he sacrificed a cow..." The angels record those who attend, the first then the next, until the Imam ascends the minbar. At that point, "they fold their scrolls and come to listen to the remembrance." So, whoever enters the mosque before the Khatib ascends is included in the virtue of early arrival and is recorded in the angels' scrolls, and the earlier he is, the greater his reward. As for one who only arrives after the Khatib has ascended, he is not included in the angels' recording of the early comers, but Jumu'ah is still sufficient for him, and he is rewarded for the remembrance, supplication, and prayer he attends, and the guardian angels record him.