The solution to this question mentions the difference in degrees of freedom for the two model to be 2. My understanding is that this should be 1 based on the following logic: Model 1 has 2 parameters and model 2 has 3 parameters, assuming they are built on the same number of observations, i.e., 4, the difference in parameters to estimate the degrees of freedom for the chi square statistics should be 1 and not 2. Is there something incorrect in the solution provided or am I missing something?

I think you are confusing the number of parameters with the number of factors. In any case, it is not the number of factors that really matters but the number of levels of each factor as that ultimately determines how many parameters you will have in your model. The difference between Model 1 and Model 2 is that you are removing one factor with three levels. For a factor with three levels you will need two parameters in your model. Thus by removing this factor, you are removing two parameters and hence the degrees of freedom changes by two.