Q:

When to use «раз, два, три» instead of «один, два, три»?

I know that один, два, три is equivalent to one, two, three in English. But, I've also heard раз, два, три used in multiple Russian songs from different artists, and I've seen this in several places online.
German has zwei & zwo for the number two, with zwo used primarily to make the #2 more easily distinguishable from the #3 (drei) when multiple numbers are being communicated simultaneously.
However, it's not clear to me whether this is the case with раз. I wouldn't expect that there's much of an issue with people confusing один and два for one another.
So, I have the following questions:

When should раз be used instead of один (and vice versa)?
What are the rules for the usage of раз?

A:

Один, два, три is usually used when counting items, things. 
Раз, два, три is usually used when counting events (say, time in seconds, ticks) or actions. 
In the later case one can interpret it as being a contraction of
[один] раз, два [раза], три [раза] 
etc, literally [one] time, two [times], three [times]. So counting objects rather than actions starting with раз is somewhat incorrect, yet sometemes used colloquially. Counting actions rather than items starting with один also sounds weird although if the action has masculine gender (like in прыжок, отжим), one can assume один be related to it.

A:

РАЗ, -а (-у); мн. разы, раз; м. I. числ. колич.
  =Один (при счёте). Раз, два, три... Раз, два и готово (о том, что делается, совершается очень быстро). Раз, два и обчёлся (об очень
  малом количестве кого-, чего-л.). II. в зн. сущ.

(Большой толковый словарь)
Instead of «один» it's used only when counting. When you are counting objects aloud you will normally say «раз, два, три, …». While «один, два три, …» is also perfectly fine (though, sounds kinda officially), «раз» is used more often.
I'm not sure what is the reason, but my guess is that it is simply shorter («два» and «три» both have just one syllable, but «один» has two, so it's more difficult to pronounce in everyday life).

