"This word has a lot of meanings."
I've searched, and it looks like the の is optional.
Here https://hinative.com/ja/questions/3380904 they say you can do that because takusan can be used like an adverb.
Jisho https://jisho.org/search/%E3%81%9F%E3%81%8F%E3%81%95%E3%82%93 defines takusan as No-adjective, Na-adjective, Noun, Adverbial noun.
So I believe it's true that the の is optional.
言葉 (ことば) can mean a word, words, a sentence, even a language. For example 彼の言葉を覚えています。(I remember his words = what he said.)
単語 (たんご) is more clear "word" but of course, it can also mean many words. For example, 単語が多すぎてわからない。(There are too many words, I don't get it.)
語彙 (ごい) is a very sophisticated word for "word" and I don't think children would understand it. The second kanji is very difficult too. However, it is still used sometimes in textbooks or so.