To your last point, there would very much be a difference in saying, "We eat in diverse places" vs. saying, "We eat in different places."
The first says you eat all kinds of different foods, while the second only suggests that at best, but could also just mean you eat at different pizza places.
Here, the speaker is not talking about specific places; they are speaking in general. This is why «nei» («in» + definite article «i») is not used; the definite article is only used when talking about specifics. The speaker is not saying "We eat in the various places [previously mentioned];" (s)he is saying "We eat in various places."