“Average” is an ambiguous term. The mode, median, and the various (harmonic, geometric, arithmetic) means are all averages.
In income statistics specifically, the usual average referenced is the median of the appropriate measure.
In context-neutral informal conversation, the arithmetic mean is the most common average.
If you want to be clear, its a good idea to avoid “average” and use the specific term for a measure.
They aren't synonyms, they are more specific terms.
In colloquial language, an average is a single number taken as representative of a list of numbers. Different concepts of average are used in different contexts. Often "average" refers to the arithmetic mean, the sum of the numbers divided by how many numbers are being averaged. In statistics, mean, median, and mode are all known as measures of central tendency, and in colloquial usage any of these might be called an average value.
https://en.m.wikipedia.org/wiki/Average
Average
1a: a single value (such as a mean, mode, or median) that summarizes or represents the general significance of a set of unequal values
Further, if you go into the definitions of median, mean, and mode on Merriam Webster's, only "mean" actually mentions the word "average." It shows up in the "synonyms" section for median, hilariously, but not in the definition itself. It is then probably least incorrect to assume that average refers to the mean unless there is some previous understanding that it doesn't (and since you presented the other meanings within a hypothetical, obviously there was no such understanding).
In any case, since I brought up the word "average" in the first place, and in all reasonability meant it to refer to the mean, I don't know why you're still arguing with me about this. You read an marginal meaning into my text to make a pedantic point and have wasted both our times.