Is the Caspian Sea a lake, a sea or an ocean?

It is a lake or a sea.

"The Caspian Sea is the largest enclosed body of water on Earth by area, variously classed as the world's largest lake or a full-fledged sea."

https://en.wikipedia.org/wiki/Caspian_sea

This variation is apparently due the lack of clear definition of a sea:

"The term sea refers to certain large amounts of water, but there is inconsistency as to its precise definition and application. Most commonly, a sea may refer to a large expanse of saline water connected with an ocean, but it is also used sometimes for a large saline lake that lacks a natural outlet, e.g. the Caspian Sea. Colloquially, the term is used as a synonym for ocean. Additionally, large lakes, such as the Great Lakes of North America, are occasionally referred to as 'inland seas'".

https://en.wikipedia.org/wiki/Sea

Additionally:

The Caspian Sea is the largest lake in the world by volume. It is also thought to be one of only a handful of ancient lakes on earth.

It is generally believed that about 90 percent of all the world's sturgeons swim in the Caspian Sea. Seven species or subspecies of sturgeon populate the Caspian, giving it the greatest diversity of sturgeon in the world.

The Caspian Sea is an endorheic, or terminal lake, meaning that its water does not reach the ocean. Because of this, minerals build up in the water as it evaporates and the lake usually becomes saline. Endorheic lakes are usually more sensitive to pollution than waterbodies that drain to the ocean.

http://www.worldlakes.org/lakedetails.asp?lakeid=8762

https://en.wikipedia.org/wiki/Endorheic

The Caspian Sea is most often considered the largest lake of the world. But since it can also be called a sea it is somewhat disputed whether it deserves this #1 position.

See also this list.
https://en.wikipedia.org/wiki/List_of_lakes_by_area

As rb6359 posts, it appears they are often used interchangeably. I'm not finding anything really clear cut:

LAKE -- An inland body of water, small to moderately large in size, with its surface exposed to the atmosphere. Most lakes fill depressions below the zone of saturation in the surrounding soil and rock materials. Generically speaking, all bodies of water of this type are lakes, although small lakes usually are called ponds, tarns (in mountains), and less frequently pools or meres.

geography.howstuffworks.com/oceans-and-seas/the-caspian-sea.htm no longer exits

SEA -- b.A relatively large body of salt water completely or partially enclosed by land. c.A relatively large landlocked body of fresh water.

answers.com/topic/sea no longer exists

Lake -- a body of water or other liquid, but usually freshwater, of considerable size contained on a body of land.
Sea --- a large expanse of saline water connected with an ocean, or a large, usually saline, lake that lacks a natural outlet such as the Caspian Sea and the Dead Sea. In common usage, often synonymous to ocean.

http://www.liquisearch.com/body_of_water/types

Salinity is a factor here, as you point out.

https://en.wikipedia.org/wiki/Salinity#Systems_of_classification_of_water_bodies_based_upon_salinity

By the way, Britannica defines the Caspian Sea as a "salt lake".

http://www.answers.com/topic/caspian-sea