Does linear kernel make SVM a linear model?

I have deleloped several SVR models for my case study using the linear kernel, and those models were optimized using the RMSE as criterion. Now Im searching for additional evaluation metrics and it turns the most publications use R squared to compare model performance during training and validation phases. It's generally suggested to avoid to use R-squared to assess the model if it uses non-linear kernel such as polynominal or radial basis function. And this refers to the fact that using R-squared for non-linear models may lead to wrong judgements.

Nevertheless, the linear kernel function equation

(1): $\displaystyle{K(x,x_i) = \sum_i x \cdot x_i}$

still looks quite nonlinear, doesnt it?

Could anybody please then explain why SVR/SVMs with linear kernel are considered as linear models, and if using R-squared for the evaluation in these cases is valid.