As Rafael suggested, "flavor" in that context means they want to see different variations of the logo.
In other words, they just want to see a set of drafts, a proof with a few suggestions for the logos! Nothing special, just another way to ask you to prepare more than one logo to choose from.
You should simply create a few different drafts of logos to choose from and ask for their feedback. Then tweak the ones they prefer until they select one. They don't mean to prepare the same logo with different variations; they probably mean different suggestions of different logo design.
At the first drafts, you don't need to spend too much time showing the logo with different variations of the same logo applied in different situations; you can simply present 2-5 drafts, and a black version for each of them as well. Later you can show them the logo in other contexts once it's closer to be final.... to help them select the best one.
Of course if you have a good budget for this, you can prepare many logos and show different variations of each but from experience I can tell you it's better to not show too many!