Require, request and requisition

Question

According to their dictionary definitions the principal meaning (as verbs) would be to ask, to demand.

To requisition is a little bit more complicated (may take an infinitive) which is why I think, it is more likely to be used as a noun next to “make” i.e. “to make a requisition”.

As for the verbs to require and to request they seem to be closer and can be used to ask and demand something as follows:

I require a new car.

I request for a new car.

Which would be the nuance differences between them regarding the colloquial, formal or polite usage?

As a noun I know and can understand the differences:

A product request refers to its demand on the market while the product requirements refer to the conditions that the product must comply with.

Answer 1

They are different. Firstly, let's see all three verbs in concern.

The verb requisition is used more for official purpose.

requisition - Make a formal request for official services and also meaning #2

An example from OALD for this - The school was requisitioned as a military hospital.

The verb require has several meanings but in the sense you are guessing, it means request and expect (#2)

An example from WW for this - We require our secretary to be on time

The verb request is to politely express that you need something

request - Politely or formally ask for

An example from OxfordDictionaries for this - the chairman requested that the reports be considered

Now your examples:

I require a new car over I request for a new car

Obviously, the latter sentence is more polite as it includes the word request.

Is there any difference in these both -- yes, there is.

The former one is just a statement that shows your requirement. On the other hand, the latter one shows you are politely asking for that requirement. To be a bit straightforward, I require a new car is ordering someone over I request for a new car is asking for a favor and doing so, that someone may oblige you by getting you a new car and not old! :)