Definition: honest distance function What does honest distance function mean?  In the context of metric spaces. 

This is not a mathematical term but a figure of speech. One says that A is an honest B to assert that A satisfies the definition of B. Usually the reason to emphasize honest is that A was informally called "B" earlier in the text.  This is associated with a somewhat conversational style of writing.
Specifically for distance functions, this word would emphasize that all axioms of metric are satisfied.
Three examples I made up: 

Given a set $E$, define its counting measure  $\nu(E) $ as the number of elements of $E$, or $\infty$ if $E$ is infinite. Next, we're  going to check that $\nu$ is an honest measure.   

and

Given two nonempty compact sets $A,B$, define the Hausdorff distance $d_H(A,B)$ as the infimum of $\rho$ such that $A\subset B_\rho$ and $B\subset A_\rho$.  To show that $d$ is an honest distance function, the only nontrivial property to check is the triangle inequality. 

and finally

Given two nonempty compact sets $A,B$, consider the minimal distance $D(A,B) = \min_{a,b} d(a,b)$. Despite its name, $D$ is not  an honest distance function: it fails the axioms of positivity and triangle inequality.

