It’s not really such a different meaning, it’s just shorthand for something like:

    X’ = X + 1
    DELETE X
    X = X’

Even if you treat it as “storing a value in a slot”, a comprehensive model of the operation needs to model the fact that the value of the slot changes over time. So another way to look at it would be:

    X_t1 = X_t0 + 1

…except that “X” is shorthand for “the latest binding that uses this name.”