I would prefer no reference to categorization, strictly speaking. I'm highlighting the points where it makes no sense to even use "atonality" as a category. Music is a cultural phenomenon, this much is clear. We should have more anthropological/ethno-musicological terminology associated with our discussions of music, in my opinion. This removes the propensity of inequity in the way we discuss and view music. We tend not to run as much of a risk in creating preferential treatment for one style of music over others. So, in the sense that we discuss "tonal" music, we should be referring more specifically to Western Classicism. When we discuss "atonal" music, I prefer to think of it as music that is "Post-Classicism" or literally "after Classicism." The sense that these styles are different has much more to do with social, philosophical, and/or aesthetic differences within culture, not the mechanical differences in the styles.
To say you've written "an atonal piece" is to say you've formed an appreciation for the philosophical, cultural, and/or aesthetic values of the Post-Classical styles of music emerging in the 20th Century, an appreciation that likely influenced your decision-making process in composing the work. Similarly, an appreciation of Western Classicism of the 16th to 19th Centuries will also share the common trait of influencing your decision-making process in composing music. Any and all styles are simply emergent systems of a musical vocabulary and syntax that incorporates a broad gamut of other influences. We can be more specific in identifying "what" from Classicism influences our music, "what" from Post-Classicism influences our music, etc. This is a much more equitable and specific way of understanding music that doesn't require us to "take sides" as though we're vying for some kind of "fan-based" popularity when we think about music. There's no reason for making mechanical, objective distinctions among styles when those observations exclude the social, philosophical, and aesthetic fundamentals on which such decisions are made.
I think the confusion here on whether or not your piece is "atonal" is based on a misunderstanding of aesthetics. Remember that while I think the Semantics are useless, accuracy is important. We just have to get past our misconceptions that emerge from an insufficient context and form a higher understanding based on practical application. You hear "atonal" and "tonal" thrown around in theoretical discourse, and you connect the terminology to the "mechanics" of the style being referenced instead of the "social, philosophical, and aesthetic" concepts that influence the style. Therefore, you're fundamentally misinformed by A) insufficient use of terminology, B) an improper context, and C) perpetual inconsistencies that emerge from insufficient terminology and context.
I have issues with the presentation of SSC's statement, but I see no sense in baiting a response from him on whether or not "tonal center" exists. This is an aesthetic issue where interpretations upon hearing music are as diverse as the opinions of the people listening. Everyone is entitled to their own view.
Take some time to review Schoenberg's Harmonielehre where he specifically advocates for the advancement of music by eliminating the barriers imposed on creativity by the functional mindset of harmony in Western Classicism. That's a good place to start, at least. You should find Schoenberg to be articulate and well-spoken on the subject, even if one might disagree with his views on the matter.