What is the definition of visibility as seen in aviation weather reports?
All my quotes for this post comes from FAA Order 7900.5C, the Surface Weather Observing Guide.

Visibility is defined as "Visibility is a measure of the horizontal opacity of the atmosphere at the point of observation and is expressed in terms of the horizontal distance at which a person should be able to see and identify specific objects."

All METARs report Prevailing Visibility, which is: "Prevailing visibility is the greatest visibility equaled or exceeded throughout at least half the horizon circle, which does not necessarily have to be continuous. This is the visibility that is considered representative of visibility conditions at the station."

Automated stations, generally report a max visibility of 10 SM. Now if you look at stations that are completely manual, i.e. KBJC(when I worked there it was a completely manual station), we'd report anywhere up to 70SM or higher visibility, due to our visibility markers.

-RA can often be easily seen through, RA or RA+ do become harder to see through, and as always, it can be dependant on where the -RA is, if it's only in a tiny quadrant of the horizon circle, it can often be a greater visibility.

[Edit Post comment about RA]: Sorry, used a few more abbreviations. SM = Statute Miles, which is like a regular mile that most people use in their daily lives. It's depicted as SM, due to most other things in aviation being in Nautical Miles.

-RA, RA, +RA, are the various descriptors for Rain. -RA means light rain, which if just using vis concerns is defined as, "From scattered drops that, regardless of duration, do not completely wet an exposed surface up to a condition where individual drops are easily seen." RA, is Moderate Rain, defined as: "Individual drops are not clearly identifiable; spray is observable just above pavements and other hard surfaces" and +RA is heavy Rain, "Rain seemingly falls in sheets; individual drops are not identifiable; heavy spray to height of several inches is observed over hard surfaces."

Also, METAR is the type of aviation weather report. METAR's are the hourly/standard report, and then there's SPECI's which if certain changes occur in conditions, from Freezing Rain, to changes in visibility or cloud ceilings, to massive wind shifts, to thunderstorms, to other factors, can be produced outside of the hourly/regular report.