Definition of average power?
The first definition works for deterministic as well as for random signals. For random signals we define the autocorrelation by

$$R_x(\tau)=E\{x^*(t)x(t+\tau)\}\tag{1}$$

where $E\{\cdot\}$ is the expectation operator. For deterministic power signals (i.e., signals with finite but non-zero power and infinite energy) we define

$$R_x(\tau)=\lim_{T\to\infty}\frac{1}{2T}\int^{T}_{-T}x^*(t)x(t+\tau)dt\tag{2}$$

From $(2)$ it is clear that for deterministic power signals both definitions of average power are equivalent, because for random signals as well as for deterministic power signals we have

$$S_x(f)=\mathcal{F}\{R_x(\tau)\}\tag{3}$$

Also take a look at this related answer.

You are right about the power of white noise. In continuous time, white noise has infinite power. Only discrete-time white noise has finite power. In sum, teachers and professors are only humans.