The Gram matrix (or Gramian) of a set of vectors in an inner product space is defined as the Hermitian matrix of inner products, with entries given by $G_{ij} = \langle \mathbf{v}_i, \mathbf{v}_j \rangle$. When these vectors are the columns of a matrix $X$ and the vector coordinates are real numbers, the Gram matrix simplifies to $X^T X$, as per the traditional definition.
Recently, I've come across instances where matrices of the form $X^T Y X$ are referred to as Gram matrices, with $Y$ being a positive definite matrix. This variant seems to introduce a weighted inner product, diverging from the traditional form where $Y$ is absent.
Considering the original definition of a Gram matrix, does the inclusion of a positive definite matrix $Y$ in the form $X^T Y X$ stretch the definition too far, or is it an acceptable extension within certain contexts? I'm trying to understand whether calling $X^T Y X$ a Gram matrix is a misuse of terminology or if there's a broader interpretation that accommodates such forms.