Exercising Stock Options Math (help)

For the life of me, I can't understand how this math is meant to work, so I'm hoping someone can explain it me.

So here's the scenario: I have received stock options from work – let's say I can exercise 10 shares at $20 per share. Cool, we're starting at $200 to purchase. However! I needed to wait a year for the options to vest, and even though I can buy 10 for $20, the current stock price is $30. It looks like I'm getting taxed on that $10 difference as an additional fee (this is via eTrade, in case that's relevant).

2 years later, the stock price is now $50 a share and I decide to sell my 10 shares for a “profit” of $300 ($50 – $20)*10, and then come April those $300 get taxed as long-term capital gains.

Here is my question – it certainly looks like I was taxed twice – once for hypothetical, unrealized gains when first exercising those stock options, and then taxed a second time on actual gains when those stocks were later sold. Is that accurate? What is going on here?

Thanks for the help.