M is for mega, k is for kilo. See for example http://physics.nist.gov/cuu/Units/prefixes.html. And these are Greek rather than Roman. Thus, a Roman M is indeed a thousand, and a Greek M is a million.
There is also the dualism of decimal vs. binary quantities, e.g. is kilo 1000 or 1024. It can be either.
I've revised and extended my original answer.
In common usage, a cardinal number is a number used in counting (a
counting number), such as 1, 2, 3, .... Weisstein, Eric W. "Cardinal
Number." From MathWorld--A Wolfram Web Resource.
http://mathworld.wolfram.com/CardinalNumber.html
… [A]n ordinal number is an adjective which describes the numerical
position of an object, e.g., first, second, third, etc. (Weisstein,
Eric W. "Ordinal Number." From MathWorld--A Wolfram Web Resource.
http://mathworld.wolfram.com/OrdinalNumber.html)
The number, “three", is cardinal and "third" is ordinal. Nonetheless, in the sentence, "He was number three to finish the race", ordinal numbers merit some attention.
Imagine a collection of runners at the finish line, discussing the results of the race.
One runner says, " I was number three to finish the race".
Another says, "I was number two."
Still another runner says, "I was number one"
These runners are discussing their ranks, that is, the order in which they crossed the finish line. They are not discussing how many runners are in a set.
Cardinal numbers are being employed to assign positions to runners along an ordering. This is possible because the cardinal numbers are themselves ordered.
For the case at hand, the cardinal number three is being used as a “figure of speech” (https://en.wikipedia.org/wiki/Figure_of_speech) for the ordinal number, “third”.
Best Answer
I don't think there is a universally accepted formal answer to this. This answer is my own style, which is reflected by Wikipedia.
From Wikipedia's article on the numero sign,
Use № only when you are using numerals (not when the number is spelled out), and only when you are referring to the number as an identifier/designation rather than as a quantity.
So I would advocate for:
I would advocate against:
You also brought up using numero signs in tables. It is considered acceptable to bend style guidelines when it comes to layout and other graphic design; if "№" fits where "number" does not, feel free to use the shorter version.
(There is also an archaic usage for the numero, in which it may be used to indicate quantity, as "№" does have a definition in which it means "in number," but I've never seen this usage in practice.)