It's an idiom. "Playing god" is a universally (or at least globally) understood figure of speech. "Jumping the shark," "letting the cat out of the bag," "messiah complex," these things apply whether or not the person being referred to has any personal notion or connection to them or not.
But NONE of that is relevant. Whether or not Catherine Halsey believes in any form of divine being (to my knowledge neither a confirmation nor a denial of this has EVER been stated, explicitly or implicitly, ANYWHERE in the Halo fiction) means absolutely nothing. The simple fact that she feels that she is smart enough, wise enough, skilled enough, or whatever enough to determine the proper and most beneficial direction the human race should take, and then implements it according to her will and design, THAT is "playing god" whether you're a devotee or atheist. Being that she refers to her Spartans as the next best step in human evolution it can be safely assumed she at least beliefs in evolution, at the very least "micro-evolution," that is the adaptation of a species resulting in subtle changes as opposed to "macro-evolution," which involves a change in species (amoebas becoming fish becoming amphibians becoming land-dwelling reptiles becoming avian reptiles becoming birds, for example). Under that assumption one could surmise that perhaps she has no regard for the supernatural (a rather unscientific approach, as science should embrace all possible explanations, even those that cannot yet be reproduced in a lab by our current level of technology). Regardless, by placing herself in the position of control over the "evolution" of humanity through her choice of artificial augmentations, she is still "playing god," and that is just one of many reasons why she is perceived as a potentially dangerous threat, which is the subject of the original question at hand here as to why the UNSC/ONI put a hit out on Halsey or had her locked away in the first place.