Usage of var keyword in the core

Normally WordPress backward compatibility is very strong, (sometimes too much strong IMHO).

So it’s close to impossible that a public property (properties declared with var are public at any effects) become private in a future release. However, as you noticed, the variable is marked as private in class doc block, this mean that developers should not use that variable directly, but via public methods, so possibility that in a future versions (when core developers decide to remove any PHP 4 code) property will be declared as private increase.

What I can suggest is ignore recomendation and access that variable direclty, or better, use a getter like

function get_errors() {
  return $this->errors;
}

and if in future versions the variable became private and no public getter is added, try to adjust your code, or maybe submit a patch to core that add the getter and/or declare the variable as protected.