All perfect praise be to Allaah, The Lord of the Worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad is His slave and Messenger. We ask Allaah to exalt his mention as well as that of his family and all his companions.
The manner of performing 'Eed prayer as mentioned in the question is the method of Imaam Abu Haneefah and also according to one of the narrations of Imaam Ahmad. The methodology of these two schools is that there are six additional utterings of Takbeer (Allaahu Akbar): three in the first Rak'ah and three in the second one. The ones in the first Rak'ah are mentioned before the recitation of the Qur'an, whereas those in the second Rak'ah are mentioned after the recitation of the Qur'an. Some companions have reported this method of performing 'Eed prayer. It is most likely that the questioner has counted the first Takbeer (the Takbeer of initiating the prayer, which is called Takbeeratul-Ihraam) that is why he stated that it is four utterings of Takbeer in the first Rak'ah.
The most preponderant opinion in this matter, however, is that of Ash-Shaafi'ee school, which is that the additional counts of Takbeer are seven in the first Rak'ah other than the Takbeeratul-Ihraam and five in the second Rak'ah other than the transitional one (when standing from the first to the second Rak'ah). Al-'Ayni stated 19 different opinions about the additional utterings of Takbeer and when they are stated, and said that we understand the difference of opinion in a way that the Prophet performed 'Eed prayer in all those different forms. Though the first opinion is the most preponderant one, still there is a difference of opinion among the scholars since the time of the companions so it should not be a cause of division and dispute among the Muslims. The matter as Imaam Ahmad had stated is that the consensus of the Muslims is an absolute evidence and their difference is a vast mercy. Therefore, you can pray behind whoever you want to (of the two groups that you have stated in the question).
Allaah knows best.