From Cern's own ask an expert section of their website.
The cost for the machine alone is about 4.6 billion CHF (about 3
billion Euro). The total project cost breaks down roughly as follows:
} 4.6 billion CHF total cost of the accelerator } 1.1 billion CHF
total CERN contribution to the experiments (about 20% of the detector
costs, supported by large collaborations of institutes worldwide) }
0.26 billion CHF total contribution to computing (manpower and materials and both CERN’s and external contributions). The
experimental collaborations are individual entities, funded
independently from CERN. CERN is a member of each experiment, and
contributes to the budget of CMS and LHCb at the 20% level, 16% for
ALICE and 13% for ATLAS. TOTEM is a much smaller experiment, with a
total material cost of about 6.5 million CHF. The CERN share of this
is, however, 30%.
4.6 CHF converted to dollars is $5,054,940,000.00 according to Google's conversion tool. This number is the number that the LHC actually cost to build, and does not include the budget for operation nor the repairs done in 2008 and 2009.
Although this article from Popular Science(PopSci) states that the LHC had a construction cost of $7.82 billion. This figure may be the initial cost plus the cost of repairs that had to be done, it does not say. It also shows that the LHC has an annual operating budget of 1.2 billion (Numbers are on a picture, so I cannot direct quote them). Which if you add the 7.82 billion and add 1.2 bil 4 times for the operating cost for the years of 2008-2012, which is when they found the Higgs boson particle, you get 12.62 billion.
According to Forbes though...
The Large Hadron Collider was first turned on in August of 2008, then
stopped for repairs in September until November 2009. Taking all of
those costs into consideration, the total cost of finding the Higgs
boson ran about $13.25 billion.
The 10 bil number that you have quoted could be the combination of the initial cost, repairs, and the annual budget that it took to find the Higgs boson particle.
You have posted articles from 2008, the 4.4 bil figure, and 2010, the 10 bil figure. So at the time of the 2008 article, 4.4 bil was roughly correct depending on the currency that was used. In 2010 the 10 bil figure was roughly correct. By using the number supplied by PopSci, in 2010 the cost came to $10.22 billion.
So to sum it up, Both figures appear to be correct at the times that they were written, but neither is correct now. The total cost to find a Higgs boson particle, which was a major purpose of the LHC, appears to be between 12.62 billion, based off the numbers cited by PopSci and my own calculations, and 13.25 billion based off of Forbes calculations.