Definition of Eigenvalue for ODE

Question

I am confused about the usage of the terms eigenfunctions and eigenvalues for ODE’s. I am used to thinking of eigenvalues for some linear operator $D$ as the solutions $$D y=\lambda y$$ for some eigenfunction $y$. But I’ve seen in places that say an eigenvalue $\lambda$ satisfies something such as $$y''+ \lambda y=0,$$ but wouldn’t the eigenvalue be $- \lambda$ in this case for linear operator $$\frac{d^2}{dx^2}$$ not $$\frac{d^2}{dx^2} + \lambda$$ with eigenvalue $\lambda?$ This seems like a minor nuance, but is the second usage simply an informality that simplifies the analysis of ODE’s or is there an alternative I am missing?