If you're practicing proof in a class (where your reader is the one who grade it), then the rule-of-thumb is that you should imagine your reader to be one of your peer: someone who read through the same books and sat through the same lectures up to the current point and understood the materials, and now do not know how to solve the current problem. The second rule-of-thumb is this: write at the level of the question. If question is asking you to prove something elementary and obvious, write down to basic axioms, but if it's something hard and not obvious, you can skip more details.

That are basically rule-of-thumb for the context of writing proof in a class. Most people who learn math only write proof in this manner anyway, so it's actually more applicable than other rules. That's something I tell my students.

If you're a writing proof for a honest audience (people who are actually interested in knowing how the arguments work, or at least want to check if it's true or not), then you should write at the level of the audience. But of course, this is not a strict rule and there are huge range of preference, even among mathematicians. Some mathematicians write extremely tersely, and some includes a lot of details. So it's not like mathematicians have a common standard.

(in fact, there was a number of controversies in math world where one mathematician write too little details, others have to go and fill them in, and then argument occur whether the others should really get credits for the work)

No, these tools are not only one. The number of tools will increase as your knowledge increases. It's just hard to explain all the tools in a book for general audiences, because they would not have known the necessary knowledge to even use such tool. In fact, for the most part, the point of mathematical theorems and theories is to give you more tools (but not all tools come from theorems).