So the definition of linear regression is that the response variable is a linear function of the estimators. If we consider univariate regression (for ease of visualization), we have $$ y = \beta_1x + \beta_0 $$ But we could also have $$ y = \beta_1x^2 + \beta_0 \\ y = \beta_1 \exp(\log(x^3)) $$ which also satisfies the condition that the response variable is a linear function of the ESTIMATORs.
I find this terminology to be a bit confusing as I would expect "linear" regression to be restricted to strict lines in univariate regression.
When linear regression is introduced in courses, the examples are always straight lines, and I think some instructors even introduce linear regression as fitting a LINEAR line to a set of data, but that's not true.
So isn't it rather confusing that it's called "linear" regression? I feel like "linear" regression connotes that the fit will be a straight line (in the univariate case).