Just to point out... Tannins is actually Tannic Acid... and it's the tannic acid that makes the water more acid, or lowers the PH...
So the more Tannins the wood leaches the more effect it will have on PH, the less it leaches the less effect it will have... Though I should also add, just because your water isn't turning tea colored, doesn't mean your wood isn't leaching any tannic acid at all...
Peat will also lower PH, though it will also tea color the water...