That would be $0.03 profit, right? Is there something wrong with this logic and is there anything I need to know?
Correct would be: You buy coin/BTC for 1$ and sell coin/BTC for $1.03 (both on Binance) later... Then the profit is $0.03.
If you sell the Ether for $1.03, the profit of it is dependent from the price you paid for it earlier. Did you pay $0.80? Then the profit is $0.23.
And you can't know the profit of your BTC-trade because you'll sell it in the future.
Don't compare apples with pears (BTC with ETH).
And what you need to know: They could have trading fees