Is a 1D convolution of size $m$ with $k$ channels the same as a 2D convolution of size $m \times k$ with 1 channel? From my understanding, this is true as each element of the filter is multiplied by the corresponding element of the particular section of the data. 
If this is the case, then what is the purpose of a 1D convolution? 
If my understanding of this is incorrect, could someone please explain where my mistake is?
 A: The distinction between 1D and 2D convolutions is the number of spatial dimensions over which the kernel is convolved to produce the convolution.
If the convolution kernel sweeps over 1 dimension, it is a 1D convolution, regardless of the number of channels or the dimension of the input tensor. If the convolution sweeps over 2 dimensions, it is a 2D convolution. 
Example 1: 1D input, 1D convolution
Here's a  high-level schematic for a 1D convolution:

Incidentally, it corresponds to your example of size $m$ with $k$ channels. Each of the $k$ kernels sweeps left to right over the same input sequence, which is just a vector of scalars.
Example 2: 2D input, 1D convolution
This is also a high-level schematic for a 1D convolution:

This one corresponds to your example of a size $m \times k$ kernel with 1 channel. The single kernel sweeps left to right over the input sequence. Even though the input sequence and the kernel are actually 2-dimensional, it's still a 1D convolution. That is, if we think of the input sequence as consisting of, say, $n$ vectors of dimension $k$ stacked together, then the kernel is convolved with each patch of size $m\times k$ in the input, resulting in a a single vector of length $n - m + 1$.
Example 3: 2D convolution
Finally, for completeness, here's a 2D convolution.

This time, the kernel sweeps the $n\times n$ input left to right and top to bottom, convolving with each patch in the input of size $m\times m$, resulting in an output of size $(n - m + 1) \times (n - m + 1)$.
What gives? In example 2, although the input is 2-dimensional, the kernel sweeps/convolves only over 1 dimension. But in example 3, the kernel sweeps over both dimensions, making it a 2D convolution.
(Indeed, images are often encoded as a 3-d tensor with 2 spatial dimensions and 3 color channels, yet we still operate on them with 2D convolutions.)
