Hi Bingley,
first I hesitated, but now I think it's time that a chemist takes over.
The language of chemistry is a curious, fuzzy mix of tradition and and attempted systematics. The French chemist Pierre Laszlo has been delving into this for a while. But the following is from my own kitchen.
"Alkali" is a very old word, used for solutions of soda ash and potash (made from wood ashes) long before molecules, ionic equilibria and pH were introduced, simply based on the effect that such solutions cause: slippery skin, color change of some natural colorants e.g. red beet. "Alkali" (for carbonates and hydroxides of sodium, potassium..) and "alkaline" (for anything with pH higher than 7), also nowadays are more practice-based terms. "Base" on the other hand, is a more general and more theory-oriented term, also covering organic molecules, as long as they can "react" or "pair up" with an "acid". The trouble is that many molecules can be an acid and a base at the same time, like water for example. It entirely depends on the partner you confront it with. So "alkali" is more closely linked to the material identity, whereas "base" indicates behavior of a substance in a given situation. It only has an operational definition.
If you desire more about such curiousities, just tell me.
(By the way, I find your posts very rich and stimulating)