I don't believe ANCHOR_NONE actually sets the tooltip to the default position, it behaves essentially the same way as calling GameTooltip:ClearAllPoints()
To mimic the default behavior, what you actually want to use is GameTooltip_SetDefaultAnchor(GameTooltip, UIParent), which is defined in frameXML\GameTooltip.lua
Lua Code:
function GameTooltip_SetDefaultAnchor(tooltip, parent)
tooltip:SetOwner(parent, "ANCHOR_NONE");
tooltip:SetPoint("BOTTOMRIGHT", "UIParent", "BOTTOMRIGHT", -CONTAINER_OFFSET_X - 13, CONTAINER_OFFSET_Y);
tooltip.default = 1;
end