If they detach their tails, sorry to be a pedant, but where does the tail start?
[muttering\I will not mention trouser snake. I will not mention trouser snake. I will not mention trouser snake. I will not mention trouser snake.\muttering]
the tail starts after the vent... same as for snakes. the vent contains the cloaca, which is part of the sex organs. in snakes (not entirely sure about slow worms) the genitalia (there are 2) are just below that. if you pick up a slow worm it may well pee or poo on you, you'll be able to see the vent then. they do an amazing projectile pee as a defence.
if you look at snakes or slow worms, you will see a narrowing just after the vent... this is the start of the tail. it is very noticeable in female slow worms at the end of summer, before they give birth.
when a slow worm loses its tail it only regrows a small amount. the one in the OP has lost part of the tail. many other lizards will regrow the tails more than a slow worm, but they will not grow to the full length - a common lizard with a full tail will have a tail the same length or slightly longer than the body itself, if it loses any it will grow to about 2/3 the original. a slow worm regrowth will only involve a small amount of regrowth and mainly consists of forming a point - for reasons below.
slow worms use their tails in fascinating ways. when you find a slow worm they will often move away tail first - this is so that any attacker will assume the moving end is the head. when it is grabbed, the slow worm will drop that section of tail and scamper away.
they also have a tendency to press the tip of their tail into you if you pick them up - this led to their name of Slayer Worm as it was assumed they were trying to inject you with venom. this was then corrupted to slow worm. another theory is that they are slayers of worms.