Thank you for the comment! I added a note to the article raising the point you’ve made. I already began writing a follow up article and was thinking of putting this information there but probably better for it to go here.
To provide a bit more context, at the time of writing I didn’t realize that ImageNet contained categories for cat and dog breeds.
I construct and train the fully connected layers from scratch and for the training I utilize only the 6 training examples. To look at your comment from this perspective, the question than becomes to what extent the original convolutional layers already capture higher level information. This is something that is unclear and would be interesting to look at.