That is so very wrong. While the overall picture seems the same, the tangible difference is in what each investor actually has at the end of each transaction:
Option 1: Dividends as cash, which means cash leaving the company. Money is transferred from one entity (the company) to another (a group of shareholders). While shareholders are regarded as owners of the company, the nature of their ownership doesn't always work quite the same as say ownership of one's personal effects. Being a shareholder mostly means having certain rights regarding how one gets part of the company's profits and the right to vote on how the company is controlled. Payout of dividends means an immediate reduction of capital for the company, and an increase of income to the shareholders. So at the end the shareholders have the same number of shares and a certain amount of cash. The market value of their shares may drop at first as a reflection of the dividend payments (because technically the company's book value does drop immediately). But let's be honest here, the whole point of shares is the profit potential, so the pricing of shares needn't be in strict alignment with the book value.
Option 2: Dividends as new shares means the company holds on to capital. At the end, shareholders get more shares, which may suit them just fine if they're not in any situation that compels them to liquidate. Extra shares means having more to sell. Let's not hyper-focus on book value, since a share price needn't be constrained by that. If there's a good chance that the price per share will rise over time, then getting shares may be more profitable to an investor than getting immediate cash.
As the old adage goes, you pays your money you takes your choice. Also, it takes money to make money. But let's not get too hung-up on book value vs. share price, or on whether money truly is a "store of value". There's an awful lot of human desires and human efforts at play here, and in the end it's that interplay of desires and efforts (i.e. what efforts must one undertake to satisfy one's own temporal desires) that brings value to anything.