At what humidity does it start to rain?
Short answer: humidity is not a proxy for rain starting and no, it does not start raining automatically when 100% humidity is reached (haze or clouds can form though).  The onset of rain is dependent on many things including humidity, but a specific value of humidity is not a sufficient condition for rain.



Water vapor is a gas and invisible.  The amount of water vapor in the air can be expressed as relative humidity (RH) which is the ratio of water vapor pressure ($e$) and saturation water vapor pressure ($e_s$).  Saturation vapor pressure is the partial pressure of vapor when evaporation and condensation rates are equal, represented by RH=100%.  When RH > 100% net condensation occurs, but water has its own ideas.  

In a mixture of pure dry air and water vapor, water will not condense until around 400% RH.  Reasons for this are a bit complicated but it has to do with very small droplets being more likely to evaporate as their curvature is very large (Kelvin effect, saturation vapor pressure is higher over curved surfaces than flat ones).  Luckily for us, our atmosphere is not pure air but has small particulates suspended in it (aerosols).  Some of these aerosols are classed as cloud condensation nuclei (CCN) and enable droplet formation at lower relative humidities.  These work by forming a solute in water increasing the energy needed to break bonds and evaporate the water (Raoult's_law)

The combined interaction of these are described by Köhler theory and describe droplet growth in terms of drop size, solute and supersaturation (RH-100%).  In a nutshell, there is a critical drop size below which drop size decreases for decreasing supersaturation and above which drop size increases for decreasing supersaturation.  The critical supersaturation is the supersaturation needed to attain the critical drop size, and is generally small (e.g. 0.3% supersaturation).

Droplets below the critical size are 'haze drops' and these make up the haze you see on very humid days.  Drops that reach the critical size can continue to grow to become cloud drops.  The condensed water is carried in the air but is no longer water vapor and is not part of relative humidity (but does contribute to the parcel density)

So... when does it rain?

It rains when water vapor is in the presence of CCN, driven to a supersaturation causing growth to the critical drop size (on the order of $\mu$m) and continuing to grow to cloud drops and further to the much bigger drop sizes that make up drizzle (100-300 $\mu$m)and rain drops(mm), a process that takes around 40 minutes.  Drops will grow until the updraft can no longer support their mass and then they fall from the cloud as rain.

Your question asks at what humidity does it rain, but what surface humidity determines is how high the cloud bases are.  When the dew point depression (the difference between temperature and dew point) is high, the cloud bases will be higher than when the dew point depression is small.  As air rises it cools, and at some point 100% RH is attained.  If there is forcing for vertical ascent, parcels can rise to this height and then to a height where they freely convect due to decreased parcel density caused by the release of energy during condensation (see: CAPE).  

So far to have rain we've needed water vapor (but not at 100% at the surface), aerosols to aid condensation (CCN) and a way to cool the air to reach 100% RH via lifting.  It is these three things -- moisture, aerosols and cooling, that we need for a rain storm.  We can have 100% RH days that are just hazy or foggy that do not rain and we can have days with mextremely little RH (e.g. deserts) that result in little rainstorms or large severe storms.  We also have storms we call 'elevated convection' that are completely disconnected from surface conditions and when these storms cause rain is not related to surface humidity at all.

If you are looking for a magic trigger for rain, your closest bet will be looking at temperature, dew point and the height parcels need to attain to freely convect (LFC).  If there is forcing for parcels to get that high and instability above, then rain is a good bet.  Forcing for lift can be anything from convergence along a boundary (sea breeze, cold front, outflow from another storm), orographic lifting (mountains, hills), thermally or dynamically forced.



To address your specific concerns:


  Water vapor is the gaseous state of water and is invisible
  
  
    Invisible to me would mean that clouds are excluded from humidity.
  


Correct, clouds are not part of humidity, they are suspended liquid water drops, usually condensed onto a solute of some kind.


  Absolute humidity is the total mass of water vapor present in a given volume of air.
  
  
    That again makes me think a cloud must be included.
  


Water vapor contributes to humidity but water vapor does not include liquid water.  Cloud water, ice, snow, rain, grapple, hail all contribute to the total mass of a volume of air, but are not humidity.


  The humidity is affected by winds and by rainfall.
  
  
    Rainfall certainly decreases humidity, but it is not stated at what percentage it starts to rain.
  


Humidity will increase during a rainstorm as rain and puddles evaporate.  Temperature will decrease toward the wet bulb temperature.  As noted in the answer, there isn't a magic number of %humidity that causes rain to start.