Generally, "Internal valuation" is the valuation used by investors while the company is still private.
One way it can affect Stripe is that it makes stock options less valuable to current employees, and can influence the weight those options have in persuading new hires.
It only makes the options less valuable if they are actually offering a liquidity event, otherwise it is actually advantageous to employees as any new option grants (both new hire and refreshers) are delineated in dollars, so a lower valuation means they get more of them.
I get that this might not align with the perspective of their employees, especially if they skew young and their expectations were shaped by tech stock price dynamics of the 2010s. A lot of folks haven't yet come to terms with the new normal. From an ISO/RSU earning employee's perspective, it's better for prices to correct quickly and completely so you can start getting new grants at more reasonable valuation with real upside.
It really depends. A lower valuation also means raising money will be at lower valuations, which means investors get more of the company, which means more share dilution.
There are typically two valuations of private companies. The "internal valuation" is usually the valuation of the common shares (ie, those that are granted to employees) whereas the "external valuation" is the value of the "preferred" shares that investors purchase. The external/preferred valuation is usually higher because the preferred shares have more attractive terms (such as that you get your money back first before other equity holders are paid out).
From the article: "A 409A valuation is an independent estimate of a startup’s fair market value often used to price stock options to employees."
I don't think it's as nefarious as that. What people are calling the "public" position here is the value of preferred stock sold in a financing, and the "internal" valuation is the value of common stock. They're different things - the preferred stock has downside protection and other special rights that make it more valuable than the common stock so it should have a different price. These internal valuation reports pretty explicitly calculate the value of the common stock as a discount applied to the preferred stock price, due to the rights and liquidation preferrence and the fact that the common is not freely tradable.
One way it can affect Stripe is that it makes stock options less valuable to current employees, and can influence the weight those options have in persuading new hires.