Oil has more than one channel
Higher oil prices can support producers' revenue while increasing costs for consumers and some businesses. The overall effect depends on the cause, the duration and the exposure of the company involved.
Rates change the comparison
Interest rates affect borrowing costs and the returns available on cash and bonds. They also influence what investors are willing to pay today for profits expected further in the future.
Currency adds another layer
An investment's return in its local currency can differ from the return measured in the currency you plan to spend. A company's own currency exposure also depends on where it earns revenue and pays costs.
My way of reading a headline
First ask what changed. Then ask whose revenue, costs or financing could change. Finally, ask how much of that story might already be in the price. A plausible explanation is not proof of the exact cause of a market move.