The Oxford Learner's Dictionaries are an excellent resource.

Have got is the usual verb in <em style="color: rgb(118, 118, 118);">British English</em> to show possession, etc. in positive statements in the present tense, in negative statements and in questions:<em>They’ve got a wonderful house.</em> <em>We haven’t got a television.</em> <em>Have you got a meeting today?</em> Questions and negative statements formed with do are also common:<em>Do you have any brothers and sisters?</em> <em>We don’t have a car</em>.

In <em style="color: rgb(118, 118, 118);">North American English</em> have and forms with do/​does/​did are the usual way to show possession, etc. in positive statements, negatives and questions:<em>They have a wonderful house.</em> <em>We don’t have a television.</em> <em>Do you have a meeting today?</em> Have got is not used in questions, but is used in positive statements, especially to emphasize that somebody has one thing rather than another:<em>‘Does your brother have brown hair?’ ‘No, he’s got blond hair.’</em>

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In both <em style="color: rgb(118, 118, 118);">British English</em> and <em style="color: rgb(118, 118, 118);">North American English</em> have and forms with do/​does and did are used when you are referring to a habit or routine:<em>We don’t often have time to talk.</em>

In [AmE] speech, you have to consider the contractions.

I think speed is what drives the get usage in informal settings.

I have got a cold. (No)

I’ve got a cold. (Yes, natural, fast, common)

I have a cold. (Yes, natural, a little slower, “better”)

I’ve a cold. (No)

To simplify your equation we will eliminate the word got, because it tends to be used oddly. You will hear the word “got” spoken in more of a informal slang manner. Ex:

I got a new car.

I got to go. (meaning- I have to leave.)

As far as the verb "have" it is more commonly used in most cases if not all.

I have three dollars.

She has a puppy.

Does she have nice hair?

So, yes you’re correct with the textbook using the outdated using of “got”, but you are on the right track.

Chris, thank you for the useful link. I will use this dictionary for sure.

Thanks a lot everyone! I got it. Yeah, I thought so too, that there isn't a big difference between them.