The "heat ball" is defined as it is in the note you cited which is bases on Evans's Partial Differential Equations Chapter 2.3.
For fixed $x\in{\bf R}^n$, $t\in{\bf R}$ and $r>0$, we define $ E(x,t;r)=\left\{(y,s)\in {\bf R}^{n+1}\bigg|\; s\leqslant t,\ \dfrac{1}{(4\pi(t-s))^{n/2}}\exp\left({-\dfrac{|x-y|^2}{4(t-s)}}\right)\geqslant\frac{1}{r^n}\right\}. $
The Wikipedia article Mean-value property for the heat equation also gives a similar definition.
Note that in the definition, one should replace $s\leqslant t$ with $s. To get some ideas of what such "ball" would look like, consider $n=1$ and $ E(0,0;1)=\left\{(y,s)\in {\bf R}^{2}\bigg|\; s<0,\ \dfrac{1}{(4\pi(-s))^{1/2}}\exp\left({-\dfrac{|-y|^2}{4(-s)}}\right)\geqslant 1\right\}\\ =\left\{(y,s)\in{\bf R}^2\bigg|\; 0<-s\leqslant\frac{1}{4\pi}, y^2\leq 2s\log(-4\pi s)\right\} $ To get $-s\leqslant\frac{1}{4\pi}$ one can simply observe that $\exp\left(-\dfrac{|-y|^2}{4(-s)}\right)\leqslant 1$ for $s<0$ and thus $ \sqrt{-4\pi s}\leqslant 1. $ On the other hand, taking the logarithm of $\dfrac{1}{(4\pi(-s))^{1/2}}\exp\left({-\dfrac{|-y|^2}{4(-s)}}\right)\geqslant 1$ gives $ y^2\leq 2s\log(-4\pi s). $
The boundary of the heat ball is like this:
