Can those who commit polytheism (shirk) and innovation (bidʻah) be described as Muslims?

Bid‘ah (innovation) is defined in Islamic law as worshipping Allah in a manner not prescribed by Him, or in a way that was not practiced by the Prophet (peace be upon him) and his Rightly Guided Caliphs. There is no such thing as a "good innovation" in religion.

Bid‘ah is divided into two types: 1. Disbelief-inducing Bid‘ah (Bid‘ah Mukaffirah): This is characterized by denying a matter that is unanimously agreed upon, transmitted by multiple reliable chains (mutawatir), and known by necessity as part of the religion. Examples include denying the attributes of Allah, denying that the Qur’an is the word of Allah and claiming it is created, or denying Allah’s knowledge and actions. Ruling on the perpetrator: If the intention is to demolish the foundations of the religion and sow doubt among its adherents, then the person is a disbeliever. If the person is misguided, they are ruled a disbeliever after the evidence has been established against them. 2. Non-Disbelief-inducing Bid‘ah (Bid‘ah Ghayr Mukaffirah): This is characterized by anything that does not necessitate denying the Book (Qur’an) or what Allah sent His messengers with. Examples include delaying some prayers or delivering the sermon before the Eid prayer. Ruling on the perpetrator: They are not deemed a disbeliever, but they have committed a grave evil.

Dealing with Innovators (Ahl al-Bid‘ah): They must be invited to the truth by clarifying the truth. If stubbornness and arrogance are found, then their falsehood should be exposed. Boycotting (hajr) is dependent on the type of innovation: if it is a disbelief-inducing innovation, boycotting is obligatory. If it is a non-disbelief-inducing innovation, a decision about boycotting is suspended; if there is a benefit in it, we do it; otherwise, we avoid it.

As for Shirk (polytheism), it has two types: 1. Major Shirk (Shirk Akbar): This expels one from the fold of Islam, such as dedicating any form of worship to other than Allah, or invoking other than Allah for something that only Allah can accomplish. 2. Minor Shirk (Shirk Asghar): This does not expel one from the fold of Islam, such as swearing by other than Allah, and showing off (riya’).

Riya’ (Showing Off) and its Effect on Nullifying Worship: If it is present at the very beginning of the act of worship, then the act is invalid and rejected. If it occurs during the act of worship: If it is resisted, it does no harm. If it is indulged in, then every action stemming from riya’ is invalid. If the end of the worship is built upon its beginning, and its end becomes corrupt (e.g., prayer), then the entire act is invalidated. If the beginning of the worship is separate from its end, such that the beginning is valid without the end (e.g., charity), then what preceded the riya’ is valid, and what came after it is invalid.