Both translate to "which" in English but are interchangeable in 0% of cases.
どの
Your grammar book says どの is used with a noun. That is exactly the difference. Notice your first example:
サントスさんはどの人ですか
The question asks "which person is Santos?" as opposed to just "which is Santos?" Notice the 人 is placed after どの. This is important; you cannot remove the noun after どの:
× サントスさんはどのですか - Wrong! Do not use!
So, どの means "which object?" and never just "which?"
どれ
どれ, on the other hand, is used to ask just "which?", not "which object?" It also stands for the type of noun you're referring to. Looking at your own second example:
ミラーさんの傘はどれですか。
This phrase translates to "which is Mira's umbrella?" Notice there is no noun after どれ. You cannot add one if you wanted to. That means this sentence is grammatically incorrect:
× サントスさんはどれひとですか
That isn't correct because どれ can't be used in combination with another noun to mean "which object?" You can, however, do it this way:
○ サントスさんはどのひとですか
(By the way, in general, it is better to use だれ (who) when asking about a person rather than どれ (which), but there are exceptions, like if you're looking at a picture full of people and you're asking which is Santos.)
In addition, the differences between どの and どれ as I described can be applied to その・それ, あの・あれ, and この・これ.
それはミラーさんの傘です。- That is Mira's umbrella.
その傘はミラーさんの傘です。- That umbrella is Mira's umbrella.
