Synthesis gas and water gas are the same thing - a mixture of carbon monoxide and hydrogen used as a feedstock for chemical production.
The reason for the two names is down to the method of manufacture. The older name, water gas, is used due to the method of passing steam over hot coke. The name "synthesis gas" came about due to process of synthetic steam reforming of natural gas (methane decomposition in steam under catalysis).
Producer gas is mainly nitrogen and carbon dioxide, manufactured by passing air over red hot coal - its called "Producer" gas because it was used in the production of iron and steel.
Semi water gas is a mixture of water gas (synthesis gas) and producer gas, which is manufactured by passing a mixture of air and water vapour through red hot coke. The heat generated when producer gas is formed increases the coke temperature sufficiently that water gas can be generated (as well).