Thats why I prefer a dictionary definition, because dictionaries normally provide rational word descriptions as defined by common usage, rather than some random definition provided by Professor Z's Wacky trans-universal theory on religion.
Except that by using irreligious as a theistic position you are straying from the dictionary definition which says nothing about a person's theistic beliefs. Whether you think irreligious theists are idiotic or not they exist and your definition basically negates them.
I suppose one would have to split the definition of irreligious then. There is the irreligion that indicates complete lack of religion, and the irreligion that indicates a lack of doctrine and proper practices.
I prefer the definition that matches the root meanings of the component parts, however I'll recognize that the other side exists.
The problem is that these two definitions between them encompass every single person in the entire world. Nobody follows each and every single doctrinal requirement of their religion perfectly, and anyone who isn't religious is also irreligious.
I'm not using irreligious to describe theists, because in that context, it's meaningless. To be a theist, you are also invariably irreligious at least part of the time. To follow up, because it is possible to be irreligious and a theist, and irreligious and completely without interest in religion, it's best to separate the terms.
I would call an irreligious theist a sinner, and leave the term irreligious to indicate the meaning of it's root words.