All measured correlations are "actually present in the data". If you take two data series and calculate their correlation it would be a number. This measured (or sample) correlation is certainly real and not fake. The question is what does it represent.

Depending on which statistical method you use, the number you calculate may not be the number you're looking for, or the number you'd have gotten had you used some other method. If you don't like my use of the word "real" to denote this, feel free to substitute some other word--"representative", maybe. By "redundancy" I'm not referring to the act of analyzing the data multiple times; I'm referring to using multiple methods to do so and seeing if you get the same result each time (possibly checking with a friend or two in the process).

As an example of "not real" correlation you offered the graphs from the linked page, but I see no reason for you to declare them "not real" other than because it does not look likely to you.

No, I am declaring them "not real" because they were calculated using a statistical method widely regarded as suspect. This suspect method is known to produce correlations that are called "spurious", and my link in the grandparent comment was to this method's Wikipedia page. I'm not sure if you thought the link I provided led to the original page you linked, but as you made no mention of "spurious correlations" (the method, not the page), I thought I'd mention it again.