I often hear people say next Sunday or next Friday when they actually mean Sunday or Friday of next week, not the coming Sunday or Friday of this week though strictly speaking the coming Sunday or Friday are actually next Sunday and next Friday if today is Wednesday.

Which way is correct?
People often think that what they said is perfectly clear when it is not. "Next Sunday" is practically meaningless. Say it's Friday the 23rd. The next Sunday is the 25th. But that is simply Sunday. You could call it "this Sunday", but that doesn't help the next person you say "next Sunday" to unless they heard call the 25th "this Sunday". There is no correct way to do this. You have to explain yourself each time if you want to be sure to communicate.
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I would usually say "this Friday" or "this coming Friday". When I say "next Friday" I always mean the coming Friday. However, I am afraid whoever hears me say "next Friday" could misinterpret what I mean if there is no hard and fast rule. I always make a point of making myself clear though. However most people I come across including my children who were born and bred and have highly professional jobs in this English speaking country do the same "mistakes". They would simply say "next Friday" without further clarification, hence I suppose that must be an educated way of saying it or the majority of people happen to say in this way.
For me "next" is always 7 days after "this", whichever day it may be! Today is Wednesday the 18th:

I'm going to a barbeque (this (coming)) Sunday. (= 22nd)
I'm going to a barbeque next Sunday. (= 29th)

I'm going to a barbeque (this (coming)) Friday. (= 20th)
I'm going to a barbeque next Friday. (= 27th)

Also, in America the week starts with Sunday in case that helps clarify some issues.
I suppose you are a native English speaker living among all English speakers and you never hear anything to the contrary, so it must be correct then the 'next "weekday"' has to refer to some weekday 7 days later.

I do not think whether the week starts with Sunday has any bearing on this issue if you say next always refers to some day after 7 days.

By the way I am very surprised that you say American week starts with Sunday while the Microsoft Outlook Calendar by default starts with Monday.
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healerthe Microsoft Outlook Calendar by default starts with Monday
In the U.S. our Outlook calendar defaults to start with Sunday, but the magic of "localization".

Most of our printed calendars start with Sunday as well, unless they're designed for another country.

Funny we still call it the weekend... Emotion: indifferent
I live in Australia, so my computer systems are localized to Australia.

According to the bible, I think the sabbath is on Saturday when Jewish people keep the sabbath if I am not wrong. And the sabbath is supposed to be the last day of the week when God finished creating everything and had a rest. Isn't it confusing?
Emotion: big smile interesting and confusing indeed! Maybe we should call Sunday the weekbegin Emotion: smile
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Would it be better to say Friday week or Sunday week instead of next Friday or next Sunday where the former is clearer or more definite and leaves no room for doubt?

By the way I've just Google'd this issue, https://www.google.com.au/search?q=%22next+Sunday%22+meaning and I've got some quite different answers to yours. It is very confusing. Some of them are as follows.

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