Take a real company, one which has real assets, real customers, real products and is looking to expand. Say a trusted third party evaluates that the company is currently worth $10M and the company is looking for $5M to expand production. The company would offer a 1/3rd equity stake to new investors to raise $5M in equity.
And there lies the difference. One is a real company with tangible assets. Thus their contribution is worth $10 million. In essence they are "raising" $15 million but are contributing 2/3 of that on their own with existing resources. The investors are asked to contribute the other 1/3.
I guess it really depends on how you look at it. The best way is to view it is (Company Value = Company's Investment, Investor's Amount = Investment, CI + I = Company Valuation = Total Raised).
No once again they would be raising $5M. Words don't have meaning when you try to use an existing word and create a new definition. Language works because people agree to the same definition for the same words or expression.
In that example:
The hypothetical company is raising $5M.
The hypothetical company is being valued (possibly correctly, possibly incorrectly that is up for investors to decide) at $10M prior to the equity sale.
The hypothetical company is being valued at $10M + $5M = $15M after the equity sale.
The amount raised is still only $5M.
Also I would point out it isn't necessary for a company to have "tangible assets" of $x. A company valued at $10M doesn't have $10M in tangible assets. If you add up the value of all of google's servers, and desk chairs, and company laptops it won't come anywhere near the value of the company. Companies are generally evaluated on the net present value of future cashflow.
Note: none of this should be taken that this paper company is worth the $12M in self valuation. It isn't. It likely isn't worth 5% of that but it doesn't make your statements any more accurate.