The two are clearly compatible, as we can take $$\psi(x) = \frac{1}{x^2}e^{-c/x}$$ which gives Wien's law in the way stated by the OP. In the original paper by Wien, the author starts from a version of the law which appears to be incompatible with the one stated by Planck: $$\phi_\lambda = F(\lambda) e^{-f(\lambda)/\theta}$$ where $\theta$ is the temperature, in accordance with the custom at the time. The original article by Planck mentions a specific reference for the formula he's using: M. Thiesen. Ver. Deutsch. Phys. Ges., 1900, 2, 66. The referenced journal can be found here. M. Thiesen fails to provide a derivation but quotes Wien, despite noting that
Die hier gegebene Formulierung des Gesetzes fehlt bei Wien
which means (translation mine):
The formula given here is not present in Wien's work
I could not find the two papers referenced by M. Thiesen. However, I managed to find this letter submitted to Nature in 1948 by E.O. Hercus ("The Proof of Wien's Law"), which is a brief review of Wien's proof of Planck's statement, which apparently involves fairly complicated considerations about light in a cylindrical cavity.
I could not find Wien's referenced paper, but I managed to find another one of the references, the book "The Electron Theory of Matter" by O. Richardson, which gives a proof at pages 339-342. The proof involves a lot of geometrical considerations about light in a cylindrical box, some experimental considerations and Stefan-Boltzmann's law.
EDIT. Perhaps it is useful to understand the historic progression. By considering spherical cavities, Wien (1893-1894) proves that $$\phi_\lambda = \theta^5 \psi(\lambda \theta)$$ Or equivalently, $$\phi_\lambda = \frac{\theta^5 \lambda^5}{\lambda^5} \psi(\lambda \theta)=\frac{1}{\lambda^5} \chi(\lambda \theta)$$ where we have defined a new function $\chi(\lambda\theta) = (\lambda\theta)^5 \psi(\lambda \theta)$. The functions $\chi$ and $\psi$ are unknown. This is emphasised by Richardson (p. 343):
It is probable that the relations (23) and (24) which involve the universal undetermined functions $\phi$ and $\chi$ of the argument $\lambda T$ are as far as we can get, from such very general considerations as have been employed above.
where the author adopts a different notation (his $\phi$ is our $\psi$ and $T$ is temperature). However it should be pointed out that this is enough to establish a very important result, Wien's displacement law. Indeed differentiating $\phi_\lambda$:
$$\frac{\partial \phi_\lambda}{\partial \lambda} = \lambda^{-6} \left[\lambda \theta \chi'(\lambda \theta) -5\chi(\lambda\theta) \right]$$
so the maximum of $\phi_\lambda$ is given by setting the thing inside square brackets to zero. The relevant thing is that this quantity only depends on the product $\lambda \theta$, so if a maximum is attained, it will be at $\lambda \theta=b$ for some $b$. This is Wien's displacement law.
Later Wien (1896) tries to get an explicit formula for the function $\chi$ (or equivalently for $\psi$). And he does this via a completely different path, which is illustrated in the paper referenced by the OP. The result he gets is more specific (he employs a number of assumptions, see the paper itself) but as mentioned at the beginning is compatible with the previous result.