@sasha, WildML is not my blog. I wish I had the time and skill for that :). But I still can give it a shot at the explanation.
Once pushed a sequence through the embedding layer, you have a 2-dim tensor: (seq_len, embed_dim); we can ignore the batch size dimension here to keep it simple. Now this has the same shape as an image, so your sequence can be pushed trough a nn.Conv2d layer. However, it does not make any semantic sense to convolve over the embedding dimension.
Look up the docs 2 and check at the constructor of nn.Conv2d:
Conv2d(in_channels, out_channels, kernel_size, stride=1, padding=0, dilation=1, groups=1, bias=True, padding_mode='zeros')

Where kernel_size is either a tuple of two ints reflecting the size of the kernel, e.g., (3, 5), or a single int, say 3, which implies a square kernel (3, 3). As I said, it does not make sense to convolve over the mebedding dimension. As such you have, you have to set the kernel size like:
Conv2d(in_channels, out_channels, kernel_size=(3, embed_dim), ...)

Or any other size than 3. This snippet kernel_size=(_, embed_dim) you should see in all the examples that use nn.Conv2d for text classification. If not, I would argue the model is implemented not correctly. Note that nn.Conv2d is happily using, say, kernel_size=(4, 4). It throws no error, but it’s semantically wrong in this case of text classification.
nn.Conv1d just simplifies this a bit. The constructor looks very similar:
Conv1d(in_channels, out_channels, kernel_size, stride=1, padding=0, dilation=1, groups=1, bias=True, padding_mode='zeros')

However, kernel_size can only be single int and not a tuple of ints since you convolve over just one dimension. As such, you would create the layer like:
Conv1d(in_channels, out_channels, kernel_size=3, ...)

which here implies the kernel size of (3, embed_dim).
In short, you can use both nn.Conv2d and nn.Conv2d. The only difference is that with nnConv2d you have to be tad more careful how you define the kernel size. With nn.Conv1d you cannot simply set the kernel size incorrectly.
I hope that helps.