***I have merged the two topics about Moonbeam's split tail which is why the thread reads a bit oddly.***
Long tailed bettas do have problems with splitting tails and fins. If the split is straight edged and goes straight inwards from the edge of the fin towards the body it is physical damage rather than infection.
Bettas' fins are delicate and can tear on sharp corners. This is why the 'tights test' was developed. Run an old pair of tights over any decor and if they snag it can tear bettas' fins. And why silk plants are recommended over plastic - plastic plants have sharp edges to the leaves and often tiny bits sticking out from the leaves and stems.
Bettas can also blow their fins, though this usually presents as holes rather than splits. It is caused by flaring too much, and the main cause is the owner leaving a mirror by the tank for too long. A betta will flare at his refection, and it is considered good exercise by some to show him a mirror for a few minutes every day. Some owners overdo it.
Whatever the cause, tears will heal quite quickly. It is important to keep the tank, and water, clean as tears can become infected if the tank is neglected. And the decor should be investigated for sharp edges. These can usually be removed with sandpaper.