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I would not recommend "Taylor $k$-differentiable", because the name "Taylor" always refers to the standard polynomial made by successive derivatives at $x_0$, and because as far as we know this generalization was not in the mind of Taylor. A quite standard and self-explanatory name is: $f$ has a polynomial expansion of order $k$ at $x_0$, where "order $k$" refers to the form of the remainder, $o(|x-x_0|^k)$. If you like a shorter name linked to a mathematician, I'd go for Peano k-differentiable, which I think is also customary, since Peano (I'll hopefully add a reference later) made quite an extensive and thorough study of this object. It has a nice calculus, analogous to the calculus of derivatives, though, as soon as you want deeper results, such as e.g. the inverse mapping theorem, you need to assume the existence of this polynomial expansion everywhere, with continuously depending coefficients: but this is equivalent to being a standard $C^k$ function.