As far as I know average and mean informally are interchangeable terms.
As far as weighing, the classical Greeks already were aware of not only arithmetic, geometric and harmonic means but possibly as many as 10 different types.
As discussed in Graziani and Veronese in "How to compute a mean? The Chisini approach and its applications" Am Stat 2009:
"(Oscar) Chisini in 1929 pointed out that in a practical context a mean should
simplify the problem under investigation (by replacing several
observations by a single value) so that the overall evaluation of the
problem itself remains unchanged. Therefore, the main issue is the
specification of the invariance requirement, being a function of the
observation, that we want to remain unchanged while replacing the
observations by their mean"
The authors also write:
"The approach has a double advantage. First by discouraging any
automatic procedure it makes students understand the substance of the
problem for which a mean is required. Second, it does not require a
preliminary (and necessariy imcomplete) list of different mean
Examples of invariance requirements and resulting means listed in Table 1 include weighed arithmetic, weighed quadratic, weighed harmonic, weighed geometric, weighed power, weighed exponential.
The paper works through several easily followed practical applications including Mean Traveling Speed, Mean Interest Rate, Mean Exchange Rate and others.
Finally, the authors note that Chisini Mean does not directly address important statistics like mode and median, but generalization by A. Herzel in 1961 (A paper I've been searching for) recasts the invariance constraint as an optimization problem to handle these.