I thought that they were also usually harvested at different times through the year?
You mean that tea intended to become, say, white, is harvested at different time than tea intended to become black? I don't think that's the case. As far as I know the major difference is what you harvest, but that expresses itself as quality of the tea, not whether it is white or oolong or black. For the top teas you harvest the bud at the tip of the branch and one or two immature leaves next to it (which often look silverish because of fine hairs on these leaves), such teas are known as "tippy". Cheaper teas harvest full-grown leaves. There might well be the difference in caffeine content between the two, but it's not a green/black difference, it's a good tea vs lousy tea difference.
Darjeeling is unusual in that it has two specific harvesting seasons (called "first flush" and "second flush") but both are used to make black (well, kinda-black) tea.
White tea is harvested early and immature. Black/oolong/green is a matter of post-processing.
White tea has huge variance in caffeine across varieties. Both tails of the distribution are white.
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