In an old engineer online course (DNV Ocean), they derive the probability of local maxima of a joint gaussian process. From Equation 4.1.82:
$$ f(x,w) = \frac{1}{(2\pi)^{3/2}\varepsilon\sqrt{M_0M_2M_4}}e^{-\frac{1}{(2\varepsilon^2)}[x^2/M_0+2\sqrt{1-\varepsilon^2}\frac{xw}{M_0M_4}+w^2/M_4]}.w $$
and eluding the computation, they get the following expression of a (time localized) PDF:
$$ f(x) = \frac{\varepsilon}{\sqrt{2\pi}\sigma}e^{-\frac{1}{2}(\frac{x}{\varepsilon\sigma})^2}+\sqrt{1 - \varepsilon^2} \Phi(\frac{\sqrt{1 - \varepsilon^2}}{\varepsilon}\frac{x}{\sigma}) \frac{x}{\sigma^2} e^{-\frac{1}{2}(\frac{x}{\sigma})^2} $$
$\Phi$ being the classical erf function.
And this is what they call Rice distribution. If I follow the reference therein, it brings me to the paper
D.E.Cartwright and M.S.Longuet-Higgins, "The Statistical Distribution of the Maximal of Random Functions." Royal Society of London. Proceedings Series A. 273, 212 (1956).
And in this one, they refer to an original book of Rice:
Rice, S. O. (1944). Mathematical analysis of random noise. Bell System Technical Journal, 23(3), 282-332.
which is also cited by the first page.
I am quite confused as I only know this form of the Rice distribution (formula taken from Wikipedia):
$$ f(x\mid \nu ,\sigma )={\frac {x}{\sigma ^{2}}}\exp \left({\frac {-(x^{2}+\nu ^{2})}{2\sigma ^{2}}}\right)I_{0}\left({\frac {x\nu }{\sigma ^{2}}}\right) $$
And I think the two forms are far from equivalent... Am I right? Are there several "well-known" Rice distributions? Am I experiencing a change in terminology?