Bodysuit (in the context of swimwear) simply refers to a one-piece suit.
It is usually skintight or formfitting and can have sleeves of any length or be made like a tank top.
There are bodysuits for both men and women, but women's forms are more common.
Full Swimming Suits completely cover legs and arms.
Feet, if present, might be like socks, which is the most common case.
Some suits have gloves or simple mittens.
Some come with hoods and are often called stinger suits.
Zentai Suits cover the entire body, including the face,
which is important for people with very light sensitive skin.
You can actually see and breathe through the fabric.
For more vigorous exercise you can get them with mouth and nose opening.
The look a bit like the outfits cartoon super heroes wear.
Shorty usually refers to a one-piece garment with short sleeves and legs that only cover to thigh level.
It only provides limited sun and stinger protection and is more a fashion swimsuit.
Some have no sleeves, just a tank top style and are popular for Triahlon.
Tank suit, Leotard or simply One-piece
is probably the most common form of one-piece swimsuit,
the tank suit form is inspiration for the subsequent creation of the tank top as a mainstream article of clothing.
The name "tank suit" is derived from the term "swimming tank", an obsolete term for what is now called a swimming pool.

Tank Suit or Leotard