If one fits a linear model with just one $x_i$ then does this mean df=1-2=-1? If one fits a linear model with just one $x_i$ then does this mean that the number of degrees of freedom $=1-2=-1$?
For a linear model the degrees of freedom is:
$${\rm df}=n-k$$
where $k$ is number of coefficients. I.e. in $a+bx_i$ it's two ($a$ and $b$).
But e.g. Student $t$ is not defined for df $\le 0$.

Yes, that is a way to see it, as the system will be underdetermined in that there is in infinite number of lines you could send through that single point. There would be one and only one line to connect two points (so zero d.f.). With more observations than regressors, we may start thinking about an "optimal" line, such as the linear regression slope.
