What is meant by freezer?

The terms freezer and deep freeze are synonymous. There is no such term as "deep fridge" in British or US English, though apparently there is in Indian English… where if you search what you find are chest freezers.

This means, whatever your terminology, a 'freezer', 'deep freeze' or 'deep fridge' is not the same as an ordinary ice box inside a refrigerator.

Small domestic refrigerators have an 'ice box', really little more than an enclosed but not sealed area where the actual cooling elements of the fridge operate. This is sufficient to make ice cubes, but not cold enough for long term freezer storage.

In the EU, ice boxes & dedicated freezers have a 'star' system which tells you how capable the box is for long term storage, from one to four stars. If there are fewer than three stars, then you should consider it for short term storage only.
Ice boxes can have up to three stars, only a true dedicated freezer can have four.
If there are no stars, then don't use it for anything other than making ice cubes.

Apparently the US has no official designation or rating for consumer guidance. I cannot guess how you may be able to tell whether your ice box is safe with no ratings system at all; other than if it has a thermometer in each section.

What do the star ratings mean on my Fridge/Freezer?
* One star (-6 degrees centigrade) is perfect for making ice cubes and is sufficiently cold to keep food for three or four days.
** Two stars (-12 degrees centigrade) stores food for fifteen to twenty days.
*** Three stars (-18 degrees centigrade) will keep food safe for up to three months.
**** Four stars (below 18 degrees centigrade) means that the compartment provides the ideal conditions for freezing down fresh and pre-cooked foods

Dedicated freezers, or larger refrigerators with a dedicated, separate freezer section [accessible by a separate sealed door, & not enclosed within the fridge itself] should all be three or four stars.

Refs -
EU Regulation 1060/2010 fridge freezer labels..
Wikipedia - Refrigerator; Temperature zones and ratings

This is a note on the terminology - for the use see Tetsujin's answer.

A freezer is a food-storage device that goes below freezing (0°C). This may be

• a standalone unit
  • with a door opening forwards
  • with a lid opening upwards, also called a "chest freezer" or "deep freeze"*
• the colder compartment in a fridge-freezer (often the lower one as shown in this picture)

• A compartment inside a fridge - first you open the fridge then you open the freezer compartment. This is sometimes called an "ice-making compartment", for the lowest specifications or "ice box", though that can also be used to refer to other things (small low-spec standalone freezers, no longer common, or even unpowered insulated boxes designed to be filled with ice)

* "Deep freeze" is also used more generally to refer to freezers excluding ice making compartments. Many foods will store indefinitely in these (from a safety point of view; quality may suffer)

_{Image by Wikipedia user Sannse CC-BY-SA-3.0}

Yes, most refrigerators include a freezer compartment (either as an insulated area inside the main refrigerator compartment, or as a separately reached, generally smaller compartment), where ice is made; the term "deep fridge" and "freezer" are synonyms. There are also separate freezers available, though those are less common in households.