Both electrode meet the minimum requirement of producing a weld with a tensile strength of at least 60 ksi.
The flux covering on E6010 and E6011 have cellulose based coverings that decompose in the presence of the high temperature arc and liberate plenty of hydrogen. The hydrogen produces a deep penetrating aggressive arc.
The E6013 is a rutile base flux covering. It does not liberate as much hydrogen as the E6010 or E6011, but still enough hydrogen that it cannot be classified as low hydrogen.
The depth of fusion only has to be sufficient to ensure the weld is completely fused to the base metal.
Your welder should not be welding with "sun glasses". The lens is not dark enough and his face is exposed to the ultraviolet radiation. The poor bugger probably cannot see the weld pool after a few seconds of arc time, burning the skin (sun burn) and causing permanent damage to both skin and eyesight over time. It is little wonder the weld quality suffers.
Any tradesman needs to have the proper tools to do the job properly, quickly, and safely. This fellow has a loaded gun, but the barrel is pointed at his own head!
Best regards - Al