At what rate do these common definitions converge to the constant $e=2.718...$?

The Laurent series expansion of $\left(1+\frac{1}{n}\right)^n$ about the point $n = \infty$ gives you an idea of the error

$$ \left(1+\frac{1}{n}\right)^n = e - \frac{e}{2n} + \frac{11e}{24n^2} - \frac{7e}{16n^3} + O\left( \frac{1}{n^4} \right) $$

On the other hand

$$ \sum_{r = 1}^n \frac{1}{r!} = e + O\left(\frac{1}{n!}\right) $$

Hence clearly the summation converges faster than the limit.

Example: For $n = 10$, the limit is $2.59374...,$ whereas the sum is $2.71828...$.

In the first case, \begin{align*} \left(1 + \frac{1}{n}\right)^n &= \exp \left[n \log \left(1 + \frac{1}{n}\right)\right] = \exp \left[1 - \frac{1}{2n} + O\left(n^{-2}\right)\right] = e - \frac{e}{2n} + O(n^{-2}) \end{align*} as $n\to\infty$. In the second case, \begin{align*} \sum_{k \leq n} \frac{1}{k!} - e = \sum_{k > n} \frac{1}{k!} &= \sum_{k > 0} \frac{1}{(k + n)!} \leq \sum_{k > 0} \frac{1}{n^k\, n!} = \frac{1}{n!(n-1)} = O\left(\frac{1}{n!}\right) \end{align*}