Equations (2.3), (2.5) and (2.7) are linear models; the others are non-linear in their parameters. Note, that the variables of linear models can take non-linear forms (\(x_1^3, x_1 \cdot x_2, \log x_1\)) as long as the parameters are not implicated in this non-linearity.
We have 6 main predictors, 5+4+3+2+1=15 2-way interactions, 20 3-way interactions, 15 4-way interactions, 6 5-way interactions and 1 6-way interaction, i.e. 63 possible predictors. Note the symmetry of this calculation: the number of possibilities of combining 2 variables (2-way interactions) is the same as the number of possibilities of leaving out 2 variables (4-way interactions).