In response to the found statement:
This does not make sense. Literally, "There is a lot of water today." So since it is not sensible, i would not use it to debunk your thoughts.
Let's consider rain and the two following statements to try and express "It rained a lot today, didn't it?"
Simply, the second is nonsensical. Here you can see the clear difference between 多い being countable and たくさん as either countable (as shown in your question) or uncountable. Instead, what you could say to use 多い and rain would be: (possibly what the initial 水が多い statement was meaning.
最近雨が多いです。The frequency was high.
Also, another thing to take note of for those who are newish to this kind of grammar is that たくさん is an adverb and 多い is an adjective and so the normal rules apply. adverb + verb, adjective + be-verb (です) or noun.
Now, to clarify, I am quoting from the Japan Times "A Dictionary of Basic Japanese Grammar", pg 354. This also slightly contradicts my answer above.
多い means (of quantity or number) a lot, many, a lot of, much. Similar to 大勢、たくさん. (大勢, it is noted, is only used for people.)
Going of this and other articles, like the one from Asahi, 多い is used correctly when referring to the quantity of an uncountable material or thing (like snow, water, rain) but is not as flexible as たくさん. I will illustrate with a few samples from the text.
"Unlike the English 'many', the Japanese 多い cannot be used before a noun, except in a relative clause where 多い is the predicate of the clause, not the modifier of the head noun."
This is an example where 多い is not as flexible as たくさん.
4. 京都（に）は多いお寺があります。 Incorrect. Corrected is 京都には多くのお寺があります。
5. お寺が多い町は京都です。 Correct. (the predicate of a clause thingy)
There are also examples of where たくさん can and cannot be used in place of 多い.
6. この部屋（に）は つくえが多いだ/たくさんある。
多く can be used as a noun, but the other two cannot.
(* denotes incorrect usage)
So in a nice long winded half incorrect answer, we can see that 多い is not strictly used for countable objects, nor can たくさん simply be swapped with 多い・多く in any situation.
Just follow the simple grammar rules, now knowing these extra couple of things, and you should have no problem with using them, I think.