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\phi(p) = (x^1(\phi(p)), \ldots, x^n(\phi(p)))

$$ instead of $$

\phi(p) = (x^1(p), \ldots, x^n(p)).

$$ That doesn't sound right to me. ##\mathbb{R}^n = \mathbb{R} \times \cdots \times \mathbb{R}##, which means that an element of ##\mathbb{R}^n## is already an ##n##-tuple, so why should we define an extra coordinate system on it? If we want a different coordinate system we just pick a different chart and not the same chart with a different coordinate system for ##\mathbb{R}^n##.

Do you agree or am I missing something?