After almost 40 years in the trade, of which 30+ are as a licenced master electrician, right off i notice two NEC violations in the OP's image posted.
Even though this is single phase, and as such generallly overlooked by inspectors, NEC requirements for color coding are black-red-blue when viewed from left to right, top to bottom or front to back. The only time NEC allows a change in phasing colors is in the case of three phase, where two wires need to be exchanged on terminals to reverse a motor rotation.
The second is definately a danger issue as it can potentially create back feed to the main panel in case of a fault.
For a sub panel, neutral and ground bar's may NOT be tied together. (A main panel its permissible and in most cases requiered.)
In this case the tie bar at the bottom of the panel needs to be removed, ALL white neutral conductors must be tied under the neutral bar on photo right, while all ground wires must be connected to the isolated ground bar photo left. This is called floating the neutral.
Grounded conductor can be either solid or stranded wire, bare or insulated with green or green/w yellow stripe insulation, and must be a minimum #8 in size, though most juristictions i've worked in now require a minimum #6. Any conductor smaller than a #6 cannot be taped green, it must have an insulating coating of the proper color, or be stripped to expose bare wire. The main grounding conductor needs to be attached to a minimum 1/2 x 8' copper or copperclad ground rod or 5/8 x 8 galvinized rod. IF there is water suppled to the building, a cold water ground is also necessary as is bonding any building steel, in the case of a steel building.
That being said, in the case of a 240v welder, two power conductors and a ground are all thats needed generally to safely power and protect the equipment. The ground or equipment ground protects from shock hazard while the two current carrying conductors, (in a 240 volt application) will carry the load imbalance back to the panel for each other.
In the case of a 120v outlet, the white or neutral conductor is what carries said load imbalance back to the panel completing the circuit.