When did "Betti cohomology" come to be used the way it is today? (and how is it used) This is sort of a mixture of a math and history question. 
First the math part: thinking about it, I do not actually know how to properly use the term "Betti cohomology". I know I should, but I don't. Betti cohomology of a variety $X$ defined over a field $k\subseteq \mathbb{C}$ refers to the singular cohomology of the associated complex space $X(\mathbb{C})$. But what coefficients, integral or rational? Or even local systems?  More importantly, complex conjugation induces an involution on $X(\mathbb{C})$ and then also on the cohomology. It seems that sometimes this is part of the structure of Betti cohomology, sometimes it isn't. So this is the math part of my confusion - maybe someone can tell me how I should use the term Betti cohomology appropriately.
Next (and more seriously), assuming we have clarified how the term "Betti cohomology" is supposed to be used nowadays - how did this evolve? The german Wikipedia article claims that Poincaré coined the term "Betti numbers" for the ranks of singular homology groups because these ranks agreed with numbers Betti had defined for surfaces. So, what are the possible reasons for calling singular cohomology of the associated complex space "Betti cohomology"? Which papers were instrumental in making Betti cohomology a popular term? Can anyone shed light on the history of the terminology? 
PS: I tagged the question ag.algebraic-geometry because the "Betti cohomology" seems to be prevalently used in algebraic geometry related communities. Feel free to retag if you consider this inappropriate.
 A: Some of this stuff is explained in much detail in this paper:


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*Charles Weibel, History of Homological Algebra (1999) 


In particular, after reviewing the work of Riemman and Betti, it says:

Inspired by Betti's paper, Poincaré (1854-1912) developed a more
  correct homology theory in his landmark 1895 paper "Analysis Situs".
  [...] In honor of Enrico Betti, Poincaré defined the nth Betti
  number of V to be $b_n+1$, where $b_n$ is the size of a maximal
  independent family. Today we call $b_n$ the nth Betti number,
  because it is the dimension of the rational vector space
  $H_n(V;\mathbb{Q})$.

The reference in question is:


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*Henri Poincaré, Analysis situs (1895) Journal de l'École Polytechnique
Also, the follow-up notes mentioned in the comments are:


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*Henri Poincaré, Complément à l'Analysis Situs (1899) Rendiconti del Circolo Matematico di Palermo

*Henri Poincaré, Second complément à l'Analysis Situs (1900) Proceedings of the London Mathematical Society
There's also a moder translation of Poincaré's original paper and a total of five "supplements":


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*Henri Poincaré, Papers on Topology: Analysis Situs and Its Five Supplements (translated by John Stillwell, 2009)


As for the evolution toward the modern use of Betti cohomology, this paragraph from Weibel's survey seems relevant:

Homological algebra in the 19th century largely consisted of a gradual
  effort to define the "Betti numbers" of a (piecewise linear) manifold.
  Beginning with Riemann's notion of genus, we see the gradual
  development of numerical invariants by Riemann, Betti and Poincaré:
  the Betti numbers and Torsion coefficients of a topological space.
  Indeed, the subject did not really move beyond these numerical invariants until about 1930. And it was not concerned with anything
  except invariants of topological spaces unit about 1945.

I still don't know where the use of Betti cohomology comes from, so this is not really an answer; hopefully someone else can help. My best guess is that it is from about the same time that the concept of Weil cohomology theory. The papers in which  the classic theorems of Betti cohomology are proven never use that name, see for example:


*

*Jean-Pierre Serre, Géométrie algébrique et géométrie analytique (1956)

*Michael Artin, The étale topology of schemes (1968)
