The heat kernel as a fundamental solution

Question

From my undergraduate studies I know that a fundamental solution to a partial differential operator $P$ is a distribution $u$ such that $Pu= \delta$ (no reference to any boundary or initial condition). Now, while reading about the heat equation I see that the heat kernel is said to be a fundamental solution for this equation, defined as a distribution $u$ satisfying $Pu=0$ and $\mathbb{lim}_{t \rightarrow 0} u(t, \cdot) = \delta$ (with $P$ the heat operator). I'm confused: are there multiple concepts of "fundamental solution"?