It is difficult to give you a simple answer.

The 3 definitions mean nothing if you don'know the origin (and the process before) of the different gases.

For exemple the composition of natural gases depends on their sources and their treatments.

Roughly speaking:

Natural gas is a combustible gas mixture consisting primarily of methane and some other higher hydrocarbons.It's what usually is distributed in gas grids these days.

Producer gas is a manufactured gas produced by the gasification of coal. It consists mainly of carbon monoxide and hydrogen and was used in the public gas supply before the supply was switched to natural gas for safety and efficiency reasons.

Associated gas, as far as I know, is combustible gas which is a by-product of the exploitation of oil fields. Quite often this is simply flared off because utilization would not be economically viable.

In addition to the previous answers, associated gas is gas that is produced in association with oil production.  The gas is separated from the heavier liquids and water.  Depending upon the oil field and the degree of processing, associated gas can either be almost all CH4 or it can be a mixture of hydrocarbons made up mostly of C1 to C5's with small amounts of slightly heavier hydrocarbons.

We should clearly distinguish between 'producer gas' and 'water gas'. Both are often alternately produced by either blowing air over red-heat coke to obtain 'producer gas', while coke heats itself to white-heat (i); or blowing steam over white-heat coke to generate 'water gas', while coke cools itself to red-heat (ii). The first process generates heat; the second consumes heat. The corresponding reactions can be given as follows, where (4N2 + O2) represents air according to its stoichiometric ratio: (i) 4N2 + O2 + 2C → 4N2 + 2CO, being this gas mixture named 'producer gas'; (ii) H2O + C → H2 + CO, where the obtained gas mixture is called 'water gas'. This alternate gas production sequence constitutes a coal gasification process.

I have a lot of details about this topic.

if you still need to know more, let me know.