When is a proof or definition formal? 
When is a proof or definition formal?

I've been searching for an explanation of when or proof or definition is formal. 
Sometimes, authors call their proofs or definitions informal without further explanation.
I've an idea that a formal proof is derived directly from the axioms (with references to already proven statements), or are the requirements less strict?
When is a definition formal? May words whose meaning follow only from the context be used in a formal definition, or should each use of a word be explicitly defined?

An formal statement/proof assumes that the problem structure is "sufficiently regular" for the conclusion to hold, and merely explains how to derive the conclusion assuming the unstated assumptions hold true.
Example of an informal statement:

The sum of an infinite geometric series with ratio $r$ is $$\sum_{k=0}^\infty r^k = \frac{1}{1-r}$$

This statement is informal since it is unclear whether this definition applies to some $r$ or all $r$; it does not explain whether it is intended to ascribe a sum to divergent series (sometimes possible).
A formal statement/proof is one that states all assumptions necessary for its conclusions to hold, except perhaps those that are blatantly obvious to the average audience.
Example of a formal statement:

The sum of an infinite geometric series with ratio $r$ is $$\sum_{k=0}^\infty r^k = \frac{1}{1-r}$$
  if $|r|<1$; otherwise, we say the series diverges and does not have a sum.

This statement is formal since it explains its required assumptions ($|r|<1$) and does not leave the reader guessing as to whether $1+2+4+\ldots$ was intended to have a sum.
Nevertheless, neither statement is considered incorrect; both of them are correct with reasonable assumptions.
