When I search for the definition about fixed-effects, random-effects or mixed-effects model here or elsewhere on the internet, there are a lot of discrepances. My first exposure to linear mixed-effects model was in longitudinal data analysis in Biostatistics. The definition is clear to me that the fixed-effect is the population-averaged effect, and random-effects is the subject-specific effect. Then the mixed-effects model is the model that contains both fixed-effects and random-effects. The mixed-effects model is usually the random effects model because it contains at least one fixed-effects parameters. Like time slope, you have one mean slope for all individuals in the data, and random-effects are those subject-specific slope deviating from the mean slope.
However in Econometrics, the deveoplments of fixed-effects and random-effects models have distinct definitions, which is whether heterogeneity correlates or not with the error term. Some statistical tests were developed to test whether fixed-effects or random-effects model should be used. There are lot of social science analyses adopting the Econometric approach as well. Therefore when I read the discussions about fixed-effects, random-effects or mixed-effects models posted by people from different areas, they always confuse me. Even though sometimes the mathematical defintions are similar, the modelling process and consideration behind it are quite different.
I hope there are some general discussions on Statistics and Econometrics about their discrepance in defintions or concepts rather than methodologies or algorithms used.