In Rammstein's "Moskau" the singer uses "раз два три" which I had understood as meaning "one, two, three", however I am now learning that the proper way to count up would be "один два три". I've come across "раз" in other situations, such as "еще раз".
When is it appropriate to use "раз" as meaning "one"?
You may have come across eins in German: it's the form that's used in a counting sequence (eins, zwei, drei...), whereas otherwise the numeral one is ein.
Раз is much the same thing: it gets used instead of один in the counting sequence. The difference is that it's not mandatory, and, in fact, not standard, but in everyday language, it's virtually never один, два, три but раз, два, три. (Note that the counting-down sequence is still три, два, один; using раз here would sound incorrect.)
Now раз is also a noun, meaning "time" — in the narrow sense of "an instance" (German Mal, French fois, Spanish vez). That's what it will mean most of the times you come across the word, such as in еще раз "one more time".
Раз can also be a colloquial conjunction meaning "since", "now that", "seeing as [something has happened]".
The word один is used to tell the number of things, like один день 'one day' or один человек 'one person', but раз is used in the sequence раз два три when you count objects, like pointing your index finger at objects counting aloud 'one, two, three, four, etc.' without naming the objects you count. Also, раз is used in the army while the soldiers are marching, and the officer is counting 'one, two, three' to help them keep the marching rhythm, the same in sports when the coach counts to the sportsmen. It is this last, army/sports usage that is in the Rammstein song, there are girls dressed as the Young Pioneers who used to march to their leader's counting, back in the Soviet times. Sometimes, один два три is also used for counting.
Also note, in Russian, only the word for 1 has different variants for these two purposes, but in Japanese all the numbers have two variants, one for telling the number of objects and the other for counting.
The other meaning of раз is 'a time', like пять раз 'five times', or even один раз 'one time'.
When counting, "раз" is synonymous то "один". You will find "раз" more used in military, for example, probably due to it being more concise. However, you won't usually use "раз" when counting things, where "один" is preferable:
один компьютер, два компьютера, три компьютера
Although "раз" may be used in this scenario as well (note the case agreement):
Раз дощечка, два дощечка – будет лесенка. Раз словечко, два словечко – будет песенка.
