de·bate

v. de·bat·ed, de·bat·ing, de·bates

1. To consider something; deliberate.
2. To engage in argument by discussing opposing points.
3. To engage in a formal discussion or argument. See Synonyms at discuss.
4. Obsolete To fight or quarrel.

v.tr.

1. To deliberate on; consider.
2. To dispute or argue about.
3. To discuss or argue (a question, for example) formally.
4. Obsolete To fight or argue for or over.

n.

1. A discussion involving opposing points; an argument.
2. Deliberation; consideration: passed the motion with little debate.
3. A formal contest of argumentation in which two opposing teams defend and attack a given proposition.
4. Obsolete Conflict; strife.

http://www.thefreedictionary.com/debate

ar·gu·ment

n.
1.
a. A discussion in which disagreement is expressed; a debate.
b. A quarrel; a dispute.
c. Archaic A reason or matter for dispute or contention: "sheath'd their swords for lack of argument" Shakespeare.

2.
a. A course of reasoning aimed at demonstrating truth or falsehood: presented a careful argument for extraterrestrial life.
b. A fact or statement put forth as proof or evidence; a reason: The current low mortgage rates are an argument for buying a house now.
c. A set of statements in which one follows logically as a conclusion from the others.

3.
a. A summary or short statement of the plot or subject of a literary work.
b. A topic; a subject: "You and love are still my argument" Shakespeare.

4. Logic The minor premise in a syllogism.

5. Mathematics
a. The independent variable of a function.
b. The angle of a complex number measured from the positive horizontal axis.

6. Computer Science A value used to evaluate a procedure or subroutine.

7. Linguistics In generative grammar, any of various positions occupied by a noun phrase in a sentence.

[Middle English, from Old French, from Latin argmentum, from arguere, to make clear; see argue.]

Synonyms: argument, dispute, controversy

These nouns denote discussion involving conflicting points of view. Argument stresses the advancement by each side of facts and reasons intended to persuade the other side: Emotions are seldom swayed by argument.
Dispute implies animosity: A dispute arose among union members about the terms of the new contract.
Controversy applies especially to major differences of opinion involving large groups of people: The use of nuclear power is the subject of widespread controversy.

___________

The use of 'argument' that I'm most familiar with is the notion that is used in logic (which is a little related to the computer science, math, and linguistics usages). I appreciate that more standard usage of 'argument' is 1 (a or b in particular). I guess my standard usage of 'argument' is more in line with 2, however.

When people use 'argument' and 'debate' interchangably I figure they must mean the 2nd def of argument since the entry on debate makes it clear that 'fight', 'quarrel', 'conflict', 'strife' usages are OBSOLETE (which is to say not currently accepted meanings)