They are nitpicking.
It means that he is the first to believe in and act on the revelation sent to him:
قُلْ إِنِّي أُمِرْتُ أَنْ أَكُونَ أَوَّلَ مَنْ أَسْلَمَ
Say, [O Muhammad], "Indeed, I have been commanded to be the first who submits"
— Quran 6:14
The religion of Islam has two facets: The beliefs and the shariah (laws and commands).
As far as the beliefs are concerned, they have been the same from Adam through Noah, Abraham, Moses and Jesus to Muhammad and so all of them are "Muslims": (Noah in 10:72, Abraham and his sons and Jacob and his sons in 2:130-132, Joseph in 12:101, Moses in 10:84, the Prophets of Israel in 5:44, Jesus and his disciples in 5:111).
However the shariah has differed for every ummah (nation) and "Muslim" is also the specific name of the ummah which follows the last shariah.
هُوَ سَمَّاكُمُ الْمُسْلِمِينَ
He named you Muslims
— Quran 22:78
Muhammad (ﷺ) was the first of our ummah and the first to submit to Allah's commands for this ummah, and as we say those who followed him were Khadijah, Ali, Abu Bakr, Zayed and so on.
This is the most prevalent tafsir of the verse:
أوّل المسلمين من هذه الأمة
first muslim from this ummah
— Ibn Kathir, Qurtubi, Tabari, Baghawi
There are also tafsirs that it means so in an absolute sense. In that Allah created the souls of the Prophets pre-creation and took the covenant from them starting from Muhammad (ﷺ):
وَإِذْ أَخَذْنَا مِنَ النَّبِيِّينَ مِيثَاقَهُمْ وَمِنكَ وَمِن نُّوحٍ وَإِبْرَاهِيمَ وَمُوسَىٰ وَعِيسَى ابْنِ مَرْيَمَ ۖ وَأَخَذْنَا مِنْهُم مِّيثَاقًا غَلِيظًا
And [mention, O Muhammad], when We took from the prophets their covenant and from you and from Noah and Abraham and Moses and Jesus, the son of Mary; and We took from them a solemn covenant.
— Quran 33:7
كنت أول الأنبياء في الخلق ، وآخرهم في البعث
[The Prophet said:] I am the first of the Prophets to be created, and the last one to be sent
— Tabari, Qurtubi