There are other red food colorings.
A lot of vegan and veggie lovers use beet powder for red food coloring. But as it can change the taste of whatever you are making, depending on the amount used, it isn't used often.
There are color extracts from plants of all sorts as well, but they can be expensive.
And yes, Red Velvet cake is just chocolate cake with food coloring in it. Actually, most of the bakeries I worked in used German Chocolate cake batter with red food coloring, as you don't require as much coloring on a lighter chocolate cake, but still get the chocolate taste.
The misleading name "Velvet" relates strictly to the color of the cake with the red coloring in it, not the texture or sophisticated recipe (even though its just choc cake with coloring). This is also the same for Blue Velvet Cake, Blue Suede Cake, or any other "velvet" or "suede" cake.
If you are going vegan, you might look at carob. I know chocolate is vegetable, but carob is slightly better for you than chocolate and to me, tastes better. You also need less sugar and less food coloring for carob....even though it tastes chocolatey, it isn't chocolate heavy.
Here are some interesting thoughts on making your own vegan food coloring....
http://www.yourdailyvegan.com/2016/03/homemade-vegan-food-coloring/
As for the recipe........it's all about what you want and what you like.
You can "google" Almond Flour Red Velvet Cake Recipes to see what is available.