Now, my understanding (if correct) is that this LLC is not a C-Corp,
but only being taxed as one, and it will receive an EIN.
If that previous sentence is correct, does becoming an actual C-Corp
later (f.e. the owner decides to raise funds) require getting a new
EIN?
Not quite right.
The LLC is a C-Corp for tax purposes now and remains one after the change of state law form of organization.
The LLC is not a corporation for state law purposes, however. C-corporation is a purely tax law term. An LLC is also not a corporation for the federal tax purposes of issuance of 1099s to it by third-parties.
A new EIN is not required or allowed.
It is necessary for the EIN to remain the same for tax characteristics of the original entity like amortization of formation costs, earnings and profits, inside basis of assets, outside basis of ownership interests, loss carry forwards, and the like to continue to the reorganized entity as part of the IRS tracking of those tax characteristics associated with the entity.
The conversion from being an LLC to a corporation under state law is considered a tax free reorganization of the existing corporation under 26 U.S.C. § 368(a)(1)(F), otherwise known as an "F reorganization" which is "a mere change in identity, form, or place of organization of one corporation, however effected". This form of entity conversion is tax free pursuant to 26 U.S.C. § 351. But, for tax purposes, there is continuity of entity in an F reorganization.
This F reorganization would probably be noted on the corporation's annual IRS Form 1120 (which has to be filed every year even for shell corporations that aren't funded), with a supplemental disclosure if there is no box to check for it and no separate form for it (I haven't done an F reorganization in eons and let the CPAs handle the forms involved).
There are a few reasons to convert an LLC already taxed as a C-corporation to an actual state law corporation.
As a general rule with narrow exceptions (e.g. for lawyers), third-parties do not have to issue at IRS Form 1099 information return to state law corporations that they make payments to for goods or services. Ultimately, this is just a rule, but it flows logically from the fact that either an S-corporation or a C-corporation must file IRS Form 1120 every year even if it has no activity, while a partnership must file IRS Form 1065 only in years when it has taxable activity. This rule was created before LLCs taxable as C-corporations existed and the relevant statute has not been amended to reflect the possibility of an LLC taxed as a C-corporation since doing so would not increase revenues and it impacts few taxpayers and is easily circumvented with an F reorganization by any taxpayer that cares.
State law corporations have more default rules of law regarding governance and usually more case law regarding governance as well. There are slightly more formalities to follow, but the burden involved is pretty trivial.
The default rule for state law corporations is that the shares are transferrable property so long as a public offering of the shares is not made when they are transferred. In contrast, the default rule in an LLC is that voting rights are not transferrable without the other members admitting the transferee as a member of the company, often by a unanimous vote. This makes it much easier to handle shareholder level transactions without a lawyer's involvement.
Members of an LLC will usually assume that it is taxed under Subchapter K and will therefore expect to receive an annual Schedule K-1 to IRS Form 1065 (a partnership tax return), until they learn that it is taxed as a C-corporation. Making the state law entity match the tax law rule prevents this confusion.d
One downside of the conversion is that it is easier for creditors to seize shares of a state law corporation than to seize membership interests in an LLC which is difficult or impossible (a "charging order") is the usual remedy for a creditor of an LLC membership interest.
UPDATE:
The quoted material in the question is saying that if you have an LLC taxes as a partnership or sole proprietorship and it elected to be taxed as a C-corporation (while remaining an LLC) that it needs a new EIN, which is correct.
