In my textbook it says you can use the 読みます form to talk about habitual actions, such as
But I read on the internet (source) that the correct form for this is 読んで and the sentence should be
Is my textbook wrong?
I think you're just missing what the source is referring to.
The part where it says
Is referring to this:
So it's not referring to the て form but the ている/ています construction.
So for example, 私はひまなとき本を読んでいます。
Note also that this doesn't mean that ている is used exclusively to describe a habit. Just that it can be.
No, your textbook isn't wrong, though it is giving you a more formal form of the verb to use. Colloquially, you can just use the dictionary form 読む.
As in, 私は週に三回本を読む。
The te form can be used as a request (basically, a shortened colloquial form of てください).
As in (mother to child at dinner), 野菜全部を食べて。
Does that answer your question?