While winter can be defined in different ways, and with different degrees of precision, depending on the context in which the word is used and the purpose of the definition, it, on most definitions, denotes, in the Northern hemisphere, the season that straddles the end of one calendar year and the beginning of the next one. (This answer, like the OP's question is limited to the Northern hemisphere; an analogous question could be asked, and an analogous answer given, about the use of summer for a season in the Southern hemisphere.) Because of that, if one wishes to unambiguously refer to a specific winter, one has to say something like the winter of 1998-99. The phrases such as the winter of 1998, which mention only one of the calendar years, are bound to be ambiguous (when considered by themselves) between the winter that starts in that year, and the winter that ends in it.
Some people may be tempted to make the argument that the winter of 1998 should stand for the winter of 1998-99 on the ground that the season should be referred to by the year in which it begins. Others may be tempted to argue that it should stand for the winter of 1997-98, on the ground that the season should be referred to by the year in which the greater part of it is. Neither of these arguments is, however, universally accepted, so they are not of much help in determining what somebody actually meant by the winter of 1998.
Of course, as with most other ambiguities, the context can help one to discern which winter was meant by such a phrase. For example, if one is told that something took place in the autumn and winter of 1998, it is reasonable to assume that it is the winter of 1998-99 that was intended . On the other hand, in 'the winter and spring of 1998', it is likely that the winter of 1997-98 was intended. If one finds the winter of 1998 in a context that is mostly about Christmas, chances are that the phrase stands for the winter of 1998-99, but in a context that concerns the Valentine's Day, it is likely that the phrase stands for the winter of 1997-98.
In the absence of such disambiguating contexts, the phrases like the winter of 1998 are ambiguous, and wise, precise writers will avoid them, just like they avoid the formulations such as the midnight on the 10th.