Yes, you've got it wrong. This is the baseball card view of the stock market, and it is incorrect. When you own shares in a company, you own part of the company - part of the money on the company's bank accounts and part of the company's assets belong to you.
The board, which is chosen by the shareholders, decides how to allocate these resources. Whether the profits are paid out to your bank account or stays in the company is irrelevant with regards to ownership. Part of the assets belong to you regardless.
Of course, the board can make stupid decisions, and the value of the company will fluctuate depending on investors' view of the company's future. But as a shareholder, you do own part of the company and if the board acts against your interests, shareholder laws are there to protect you.
The board, which is chosen by the shareholders, decides how to allocate these resources. Whether the profits are paid out to your bank account or stays in the company is irrelevant with regards to ownership. Part of the assets belong to you regardless.
Of course, the board can make stupid decisions, and the value of the company will fluctuate depending on investors' view of the company's future. But as a shareholder, you do own part of the company and if the board acts against your interests, shareholder laws are there to protect you.