Given a symmetric, bilinear map $B : \mathbb{R}^n \times \mathbb{R}^n \to \mathbb{R}^m$, a common notion of nondegeneracy is the following: $$ \mbox{If } B(x,y) = 0 \mbox{ for all } y, \mbox{then } x=0. $$ However, there is another notion, let's call nonsingular, defined as follows: $$ \mbox{If } B(x,y) = 0 \mbox{ then } x=0 \mbox{ or } y=0. $$ This second notion is important for immersions and embeddings of projective spaces, and so the use of the term nonsingular is common in some literature: see e.g. Section 6 of this paper by James. I would like to know more about this second notion, but cannot find much because I only see search results concerning the first notion. I am wondering

1. Is there a different name for this second notion?
2. Are there some known uses for this definition besides immersions/embeddings of projective spaces?