Multiclass classification is the problem of classifying instances into one of the more than two classes. However, the prefix multi means "more than one" (as in multi-label classification: if one example can have 2 labels, then it's multi-label classification). So strictly speaking, since two classes are more than one class, a 2-class problem (aka. binary classification) is multiclass classification. Is there any clearer term than multiclass classification?
so multi, when used in machine learning to describe a classifier is a term of art whose meaning differs from its ordinary dictionary meaning--ie, multi here refers to more than two, rather than more than one.
and indeed this usage reflects a meaningful difference among ML techniques: three is very different from two in machine learning--two classes you can separate with a single hyperplane.
this distinction is a useful one in practice as well, eg, a Support Vector Machine is, inherently a two-class classifier (the trained model is comprised of a single hyperplane). Of course is your data is comprised of ten classes rather than two, you can still use an SVM, but you have to use it differently, four separate models applied in series (class I versus all else => class II versus all else etc). That's not true for many techniques though--eg multi-layer perceptron w/ softmax, whether you want to model your data using two classes or ten, requires just changing the number of nodes in the output layer. For other techniques, that's not true, eg.,
lastly i have noticed that the terms multi-class and multi-label (both applied to ML classifiers) are sometimes used interchangeably, which is incorrect, they refer to two different kinds of problems.