Let \( x \) be the number of bins of glass bottles Lindsey collected. According to the problem, Lindsey collected \( \frac{2}{3} \) of a bin of glass bottles. Thus, we have:
\[ x = \frac{2}{3} \]
Diane collected 7 times as many bins as Lindsey, so we calculate the amount Diane collected as follows:
\[ \text{Diane's bins} = 7 \times x \]
Substituting \( x \):
\[ \text{Diane's bins} = 7 \times \frac{2}{3} \]
To perform the multiplication:
\[ \text{Diane's bins} = \frac{7 \times 2}{3} = \frac{14}{3} \]
Thus, the number of bins of bottles that Diane collected is
\[ \boxed{\frac{14}{3}} \]
This can also be expressed as a mixed number:
\[ \frac{14}{3} = 4 \frac{2}{3} \]
So Diane collected \( 4 \frac{2}{3} \) bins as well. Either \( \frac{14}{3} \) or \( 4 \frac{2}{3} \) is a valid answer.