I have a couple of comments, before answering your specific questions.
First, you might benefit by reading up a little bit on generalized linear models (e.g. on this wikipedia page). You seem to think your model is not linear, but it is. A logistic regression is a linear model -- because you use a linking function to make it so.
In particular, the word "linear" in linear regression refers to the coefficients, not the terms themselves. For example, fitting a model of the form
y = alpha + beta1*x + beta2*x.^2
would be a linear regression. It's linear in alpha and beta (not in x).
Fitting the model
is a non-linear model, because it is not linear in b.
Second, by virtue of the fact that you used fitglm, you have fit a linear model. Whether that was the most appropriate model for your data is impossible to know, without seeing your data. (That is basically the answer to your second question.)