One way to increase awareness of this issue is to explore what happens if the quantors are changed from exists to for all, and vice versa.
On math.SE, there was a question which dealt precisely with this
The following exercise is supposedly from Makarov's Selected Problems in Real Analysis:
Describe the set of functions $f: \mathbb R \rightarrow \mathbb R$ having the following properties ($\epsilon, \delta,x_1,x_2 \in \mathbb R$) :
a) $\forall \epsilon ,\qquad\qquad \exists \delta>0 ,\qquad |x_1-x_2| < \delta \Rightarrow |f(x_1)-f(x_2)|<\epsilon$
b) $\forall \epsilon >0 ,\qquad \exists \delta ,\qquad \qquad |x_1-x_2| < \delta \Rightarrow |f(x_1)-f(x_2)|<\epsilon$
c) $\forall \epsilon >0 ,\qquad \exists \delta>0 ,\qquad (x_1-x_2) < \delta \Rightarrow |f(x_1)-f(x_2)|<\epsilon$
d) $\forall \epsilon >0 ,\qquad \forall \delta>0 ,\qquad |x_1-x_2| < \delta \Rightarrow |f(x_1)-f(x_2)|<\epsilon$
e) $\forall \epsilon >0 ,\qquad \exists \delta>0 ,\qquad |x_1-x_2| < \delta \Rightarrow |f(x_1)-f(x_2)|>\epsilon$
f) $\forall \epsilon >0 ,\qquad \exists \delta>0 ,\qquad |x_1-x_2| < \epsilon \Rightarrow |f(x_1)-f(x_2)|<\delta$
g) $\forall \epsilon >0 ,\qquad \exists \delta>0 ,\qquad |f(x_1)-f(x_2)| > \epsilon \Rightarrow |x_1-x_2|> \delta$
h) $\exists \epsilon >0 ,\qquad \forall \delta>0 ,\qquad |x_1-x_2| < \delta \Rightarrow |f(x_1)-f(x_2)|<\epsilon$
i) $\forall \epsilon >0 ,\qquad \exists \delta>0 ,\qquad x_1-x_2 < \delta \Rightarrow f(x_1)-f(x_2)<\epsilon$