Is there any program or website that allows me to create phylogenetic trees without having to insert an already created dataset?
Have you ever tried using Mesquite?

I used it when I taught intro bio lab course as a TA.
Basically you try to come up with yes or no questions, and based on how you answer these questions it will generate a tree similar to your second link.

Can be a little bit complicated, but I am sure there are many tutorials for it on its website.