The question asks about an apparent ambiguity in the meaning of Christmas Eve Eve between:
(1) the evening of 24th December,
(2) 23rd December.
The OP finds it obvious that there is an ambiguity here, but seems to favour the first reading. If I understand him correctly, he regards the phrase, when used in the first sense, as pleonastic, but not defective in any other way. All four answers posted so far, however, seem to imply that, in so far as the phrase is meaningful at all, it has the second meaning, apparently on the ground that eve means the day before, and that it is applied here recursively. Everybody seems to agree that the phrase is informal, colloquial, and perhaps joking, but it is noteworthy that the four answerers all came on one side, in apparent opposition to the OP. Everybody involved is a well established contributor to this site, so neither side can be dismissed out of hand.
Chances are that different people mean different things by this phrase and that there is no definite answer as to whether (1) or (2) is the correct interpretation. Apart from the humorous uses, such as the one in DavePhD's answer, it is however not clear why anyone would ever need to use the phrase for 23rd December: it is an ordinary day and doesn't need a special name.
There are, however, circumstances in which there may be a genuine need to use it for the evening of 24th December, although even then it will be far from the best term for the purpose. To understand why this is is so, let's start with the uncontroversial fact that the original meaning of Christmas Eve was the evening of 24th December, and that this is still its primary meaning. Why do we have a special term for that evening, and not for the evenings before many other holidays? The answer is, of course, that Christians usually participate in Christmas-related religious activities at that time. Christmas Eve doesn't stand simply for the evening of 24th December, as determined by clocks, but is rather understood to mean something like the part of 24th December that is devoted to Christmas-related celebratory activities.
Traditionally, that part of 24th December was the evening, and the rest of the day was an ordinary working day. Over time, however, the boundary between the two parts of the day has been shifting: in much of the world, although the day is officially a working day, most businesses now close earlier than usual, and even during the hours when they are open, operate in a way that leaves a lot of room for celebratory spirit. That, I believe, explains why the meaning of Christmas Eve, in the minds of many people changed from the evening of 24th December to the whole day of 24th December.
The use of Christmas Eve for the whole day created a difficulty for those who wanted to refer specifically to the evening. They can, of course, continue to use Christmas Eve for that purpose and rely on the context to make it clear that it is intended in its original sense. When the context is not sufficient for the purpose (as may be the case in weather forecasts), one, however, needs to be more explicit about intending to refer to the evening. Christmas Eve Eve may be one, albeit clumsy, way of doing that. It can thus be regarded as a retronym of sorts.