I am looking to accumulate a comprehensive list of definitions for quantiles under complex sampling that have been published or implemented in software. I'm not worrying about the separate problem of uncertainty estimation.
Without the complication of complex sampling, Hyndman & Fan wrote a thorough review in 1996 Wikipedia notes that there is no standard definition in the presence of weights.
In addition to the question of how the interpolation between order statistics should be done in the presence of weights, the survey literature has also distinguished between ties due to discrete variables and ties due to rounding of continuous variables (in the latter case, the two observations bracketing the quantile might both have been rounded to the same value, and no interpolation would be needed).
