Anselmian: Bodily things cannot be the same as their nature, and so cannot have all the possible perfections of that nature. Any given human being has to be introverted or extroverted, social or autonomous, male or female, etc., while one human nature is expressed in this diversity. Therefore if God had a body, then we could think of a greater degree of perfection than God. But God is that than which nothing greater can be thought, therefore God does not have a body.
Aristotelian: When we deny that God is a body we don’t want to deny what is definite, structured, tactile, etc. but the indefinite character of the body. Take this body
We don’t want to deny its meaning, its contour, it likeness to a treble clef, etc., but only that it could be both there and here (&) or that a complete symbol-system would require more than &, or that we could turn off the computer and the symbol would vanish. We deny a body to God not so far as bodies are definite but so far as they cannot slough off this concomitant indefinite character.
Christian: We deny a body to God because, if not, then Christ is two bodies: which means either he is some mixture of two elements or that he exists in two places. But the first is the mistake of Eutyches, who posits an impossible tertium quid other than God and creature; and the second denies the Incarnation: descendit de caelis, et Incarnatus est.
Gnostic: All this is too complicated. Bodies are evil, God is good. Done.
Platonic: One only comes to know what bodies are, and therefore what is most real about bodies, by knowing what is common to all of them but limited to none. “The physical” for example, is either the extended, or the sensible, or what is given to concrete confirmation, and all of these descriptions are common to all physical things and limited to none. But no concrete physical thing is common to all but limited to none, and God is most real, etc,
Physicalist: While all that exists is physical, what is fundamental in it is abstract. Physics has always reduced the concrete to the abstract, whether to absolute space and time, or laws of nature, or idealized mathematical structures, or amplituhedrons. Since “God” is whatever is first and most fundamental in reality, God must be abstract. But no body is abstract, therefore, etc.
Spinozistic: We deny body to God because, while God is the universe, this does not mean that we can just sum up that stone, that leaf, that star, that chunk of space, etc. and get a giant heap called “God”. God is rather the substance of all things. God is a body so far as any substance is, and is not one so far as any substance is not.