Let's see if I pass the test:
An unbiased render means it uses no shortcuts (let's call optimizations?) to render a scene. It's closer to the real behavior of how light travels trough space since it traces its light samples starting from the light source. This also means way more light photons (to correlate with reality) to proper light a scene.
All renderers take shortcuts and make assumptions - and it already starts with assuming that a photon is a particle and not a wave* and using phong interpolated normals on flat triangles. Some of these introduce bias, some don't. Russian roulette for example does not introduce bias. Adaptive sampling the naive way (=adding more samples to a pixel) does introduce bias. Ignoring light sources with minimal contribution (a candle on the moon) is biased, terminating paths after a certain number of bounces or once they don't contribute significantly is biased. You'll notice that many renderers that advertise being unbiased have a "max bounces" parameter - which means they're biased. They may still be noisy progressive metropolis path tracers, but they're biased nonetheless. They're about as unbiased as a vegetable soup is vegetarian after you add two bacon bits.
Photon mapping and path tracing don't have any difference in tracing how light behaves. They can use the exact same physical models for light sources and shaders. It has nothing to do with tracing from the light source either - basic path tracing starts at the eye, not the light source, where photon mapping starts at both ends, light and eye.
Also, both a biased and an unbiased renderer can be consistent, that is eventually converge to the correct result if you let them render to infinity.
What makes a renderer unbiased then? If you let it render an infinite number of images of the same scene and compared it to reality, the average error of all images would be 0. Note that individual images can still have an error, a large one even - their average error needs to be 0. Doing the same with a biased renderer would result in an average error that's not 0, even though the error of individual images can be lower than in the unbiased version.
So really, unless you are rendering to infinity, bias vs unbiased is a purely academic exercise. Let's name things for what they are and say "path tracer" when we refer to Arnold or Maxwell and "irradiance cache" when we refer to C4D or Mental Ray.
About correctness, I dare anyone to look at these images and claim the (unbiased) path tracer is more correct than the biased progressive photon mapper:
* Don't call it physically correct until you can render doppler shift and double slit effects.