This response is not a formal definition of a multilevel main effect but it was too long for a comment. Regardless, it is included here as a hopefully useful, intuitive, non-formal discussion of the issues. With a little luck, the ruling CV participants will be flexible enough to permit such a discussion without demanding rigid fealty to the OPs precise question. If not, I have no problem deleting this response.
Main effects are most typically defined as the constant effect of one variable across all values of another variable (e.g., Aiken and West, Multiple Regression, p. 38). In classic OLS ANOVA evaluation of these effects, particularly categorical factors, rests on decisions about the presence (absence) of an intercept.
For models with an intercept -- the most common parametrization -- continuous main effects are evaluated at the conditional mean of the independent variables while dummy (0,1) variables represent distance or deviations from the conditional grand mean (the intercept), as defined by the model. For multilevel factors, the presence of an intercept necessitates introduction of a zero (0) or base level against which the other levels are evaluated in the cross-products matrix -- failure to include a zero or base level will exhaust the degrees of freedom for that factor. For models with an intercept, the factor coefficients represent distances or deviations from that pre-specified base level. The choice of which base level to use is an analyst decision, e.g., the last alphanumeric value in the factor based on EBCDIC-type rankings is one common approach.
For models without an intercept the interpretation changes as continuous main effects are now evaluated conditionally at zero wrt the other variables in the model. Dummy variables still represent the distance or deviation of that variable from the 'grand mean' conditional zero as defined by the model. For multilevel factors, the absence of an intercept eliminates the need for a zero or base level. The factor coefficients now capture distance or deviations for each level from the 'grand,' conditional zero defined by the model. 'Evaluation at zero' can lead to interpretive problems wrt nonnegative variables. Mean centering of independent variables in the case of models without an intercept is one recommended solution to this problem.