I'm having a hard time with this concept. I've read as many different sources on this terminology as I can find, but it seems that most of them summarily describe it the same way without really expounding in a manner that helps me build a conceptual understanding. I thought I might have finally started to grasp this notion, but then I fell flat on my face with the first homework question.

The model provided is: Y = 10,000 + 150*X1 + 25*X1^2 + 60*X2
The question is: Is this a “linear” regression model, why or why not?

Based on the concept as I derived it from reading, I thought it was not a linear model. Based on subsequent questions in the class work, I'm pretty darn sure that it is supposed to be linear.

I think what you need to do is understand the difference between "Linearity in the Variables" versus "Linearity in the Parameters". In your example that you provided above Y is not a "linear function" because the variable X1 appears as a power or index of 2.

That said, to be a "Linear function of the Parameters" the model Y = b0 +b1*x^2 is a "Linear Regression Model" but Y = b0 + Sqrt[b1]*X is not - i.e. the latter in an example of a non-linear (in parameters) regression model.

So "linear at the parameters" is a condition where the parameter itself (b1, b2, etc) is linear regardless of what is happening to the variables?
As I understand it, a model/function will be considered linear if the parameters are linear and even when the visual depiction of the model is not linear itself.
Putting the two together, it would be appropriate to say that the original model is linear because it is linear at the parameters (10,000/150/25/60).