can someone explain to me the point of the 'initial valuation', considering ultimately the price will settle lated thorugh trading to perceived value anyway ?
The initial buyers of the shares sold by Anthropic are institutional investors (mostly). That matching process is done by an investment bank they hire.
The trading price you see on the screen is - shortly afterwards the shares become publicly tradable on the exchange (or on all US equities exchanges, for US stocks). So in effect, the price shooting up on the public markets is actually bad for the company in that they might have left money on the table.
The price will settle, but it has to start somewhere.
Well obviously Anthropic as the one offering the shares would care about how much they sell them for.
The price those that buy the IPO get before it starts trading publicly.
E.g. SpaceX is currently trading below its IPO price. If you became a shareholder by buying at IPO and held your have lost money as of today.