Difference between "scaled white noise " vs. "unscaled white noise"
Wavelets, as bases or frames, are linear like-decompositions. As such, signals and processes can be decomposed into a linear or weighed sum of coefficients multipling (wavelet) vectors:

$$s[n] = \sum a_n e[n]$$

The indexing above is not wavelet-specific. Traditionally, there was some natural order: low $n$ denoting low-scales, or high frequencies, and vice-versa. So traditional linear filtering often consisted in keeping indices above or below some cut-off indices (with weighting), assuming that data above and below could be noise (typically low-pass filtering).

This linear (order-related) scheme remains practically useful. However, and some cases, a non-linear selection can prove more efficient. Especially when indices, across time and scale, don't exhibit clear ordering anymore.

$$s[n] = \sum a_{j,k} \psi\left[\frac{n -k2^j}{2^j}\right]$$

The main idea is to select "cleaned signal" coefficient not within indices, but with respect to magnitude: the highest the magnitude, the most likely it belongs to data instead of noise.

Selecting the highest coefficients amounts to setting a threshold below which you discard coefficients. This threshold should be selected as cleverly of possible. nd here is the catch. In the early works on wavelet thresholding, for proofs and practice, it was often considering that the noise was Gaussian with (known) unit variance. This would yield nice theories and practical algorithms.

Yet, this requires a knowledge on the noise, which is often unknown (in shape and amplitude). If we forget the shape (Gaussian or not), if the noise level is $\sigma$ (instead on one), this should be taken into account, esp. by changing the threshold accordingly. Caveat, estimating this $\sigma$  remains tricky.