Yes it is pretty cool! I learned that about a year ago.
To answer your first question: Not exactly. With Lyre tail's their two caudal fins look to have a gap that you can easily see through. With double tail's you will see no gaps. It does differ with the HMDT's and the PKDT's.
Look at these two photos and you will know from a lyre tail and a double tail. It's more of a visual than verbal that it's easier to explain.
This is the lyre tail (obviously):
Veil tail Lyre tail:
HMLT (Halfmoon Lyre Tail)
And the double tail:
Your Lyre tail would be in the category as A red and Cellophane Butterfly Veil Tail Lyretail. (Remember breeders still call lyre tails double tails. Especially to people that just want to buy.) That "clear" color on him is the Cellophane part.It is referred as a butterfly because of how the two colors separate. Though he is not a "perfect" butterfly, but he is in fact a butterfly.
This is how other "perfect" butterflies and other butterflies that don't have the entire symmetrical divide.
This is a good example of a "Perfect"
This one is also good because it is kind of like yours (Cello And red Butterfly)
The definition of a butterfly betta is when a betta has a "rainbow" like pattern in a circular form around the dorsal anal and caudal with two or more colors divided separately.