I notice that you excluded intercept from the fixed factor effects (| NOINT). When there is no intercept in a model, the dichotomous factor is not the same as dichotomous covariate. Just look at an easier example, linear model with dichotomous predictors X1 and X2:

UNIANOVA Y WITH X1 X2
  /METHOD= SSTYPE(3)
  /INTERCEPT= EXCLUDE
  /DESIGN= X1 X2
  /PRINT= PARAM.
UNIANOVA Y BY X1 WITH X2
  /METHOD= SSTYPE(3)
  /INTERCEPT= EXCLUDE
  /DESIGN=X1 X2
  /PRINT= PARAM.
In both commands, intercept is suppressed. In the first one X1 and X2 are covariates while in the second one X1 is factor. In the presense of intercept, both would have yielded identical results because one of the 2 levels of factor X1 would come out to be redundant which will actually turn X1 into a single covariate. With intercept suppressed, though, both levels of X1 are valid dummy predictors, so you actually have two covariates in place of X1, all in all 3 predictors in the model.

Thank you for the reply, this is helpful. I am just not sure which model is the correct model and whether I should include NOINT or not. I don't have any expectations that one species has a larger pupil-size than the other. That is not of interest. I am interested in the three-way interaction between species_subject, same_different, stimuluscondition, to see if the mean pupil size is different, but also if the slope is different.

Just to avoid the confusion that the question and the answer may create. In the script (SPSS) the 'noint' keyword is used to suppress the intercept, because: the "with int".... already defines an intercept. The variable int is the unit vector defined prior to the linear mixed call. This should have been emphasized. Without the "with int" the answerer would be completely right.

Thank you for your reply Conor. I edited my original question and emphasized the 'noint' and 'with int' statements. Does anybody else have an answer to my question about when to use the BY and WITH command and what the exact difference is between the two?