Both 第 (prefix) and 次 (suffix) both seem to have the effect of turning it into an ordinal (eg: "two" into "second"). And sometimes they are both used, like 第二次大戦. When does one choose one over the other, or choose both? How do they compare/contrast?
次 is a counter/unit with specific meanings. It's used in three types of situations:
- Counting or numbering phases, waves of one big phenomenon/event. 次 tends to be used for things that happen randomly but eventually finish, whereas 回 is used for regularly scheduled events and programs.
3次販売 third sale (of a popular product)
2次災害 secondary disaster
攻撃は4次に分けて行われる Attacks will be done in four waves.
- (Math) Counting a degree of a polynomial.
3次関数 cubic function
2次方程式 quadratic equation
- Saying primary, secondary, tertiary, and so on to describe the order of importance (3次 and larger are uncommon)
2次的な目的 secondary purpose
廃熱の2次利用 secondary utilization of waste heat
As for 第1/2次世界大戦, I think 次 was used because the two World Wars were considered to be parts of a larger historical process.
On the other hand, 第 is just like "no." or "-th".
第3会議室 Conference Room 3
第4開発部 Development Division IV
交響曲第5 Symphony No. 5
第3王女 Third Princess
Its primary purpose is to make an ordinal/label and avoid confusion (3会議室 would mean "three conference rooms"). 第 is common in stiff fixed phrases such as 第2次世界大戦. But since there is usually no ambiguity when 次 is present, it's often safe to drop 第; for example, 3次販売 and 第3次販売 are interchangeable.
There is no conflict, and 第二次 simply means 'the second time'. 第二 means the second and 次 means time.
Yes it's possible to say 第二大戦 to mean 'the second war', but most people would interpret it as 'the second (the inferior) war, the war that is the second in class', implying there is a 'greater war'.
Such is the habit of the Japanese (or Chinese, since this is Kango) language, to say 'the second time 第二次' instead of 'the second 第二'.