فَأَخْرَجْنَا مَنْ كَانَ فِيهَا مِنَ الْمُؤْمِنِينَ فَمَا وَجَدْنَا فِيهَا غَيْرَ بَيْتٍ مِنَ الْمُسْلِمِينَ
We brought out the believers there; and We found there only one house of Muslims. (51:35–36)
The same episode refers first to believers and then to Muslims. This pairing supports a close relationship between the two descriptions.
It does not logically prove that the words are synonymous in every usage. The more precise conclusion is that the people identified by genuine belief are also correctly identified by submission. The scope of the evidence should be neither understated nor exaggerated.