Uncovered Interest Parity question I am trying to understand the fact that, given that the uncovered interest parity holds, a rise in dollar interest rates will cause the dollar to appreciate. Is this a good intuition?

Uncovered interest parity (UIP) is defined as ${R_F=R_$}+{({E_{F/$}}^e-{E_{F/$}}^c)}/{E_{F/$}}^c$ where $$ $$ $R_F$ is the annual return of foreign currency, $$ $$ ${R_$}$ is the annual rate of return (the interest rate) for holding dollars in a bank, $$ $$ ${E_{F/$}}^e$ is the expected exchange rate (how many foreign currency units could be bought with one dollar) in one year's time, and $$ $$ ${E_{F/$}}^c$ is the current exchange rate (also per dollars). $$ $$

Example. Supposing ${R_$}$ is 4%, ${E_{F/$}}^c$ is 0.67, ${E_{F/$}}^e$ is 0.69, the rate of return for holding foreign currency $R_{F}$ is 7%. But now, a rise in dollar interest rates (${R_$}$ is now 8%) it means I fix the rate of return on dollars at 7% and see how ${E_{F/$}}^e$ changes. Is that the correct intuition? I use this approach and get that ${E_{F/$}}^e$ is 0.66, which means the foreign currency appreciated, but that means the dollars didn't appreciate, which contradicts my original claim...
Your example does not contradict the theory.

Uncovered interest parity (UIP) indicates the degree and direction of movement of exchange rates in the long term (though I see some economist supposes it's mid term). The intuition behind is that the currency of higher interest rate county has the expectation of depreciation in the future, to the percentage as the difference in the interest rates, to make arbitrage not profitable during a period of time. 

Come back to your example. Given the foreign interest rate $R_{F}$ is 7% and and the domestic interest rate $R_{$}$ is 4% (less than 7%), the domestic currency (dollar) is expected to appreciate about 3% in the future according to the model, which is consistent with your calculation (from the current exchange rate $E_{F/$}^{c}$ 0.67 to the future exchange rate $E_{F/$}^{e}$ of 0.69, a 3% change). If the domestic interest rate $R_{$}$ is 8%, while the foreign interest rate $R_{F}$ is still 7% (now domestic interested rate is higher), it is expected that the domestic currency will depreciate for 1%, which is indicated by a change from 0.67 to 0.66.

This theory does not account for short term movement of exchange rate. The underlying assumption implied by this theory is that exchange rate is determined by capital flow, which is driven solely by interest rate spread, and it does not account for market impact of rate hike as well.

In addition, a interest rate hike, according to the theory, will change the direction of appreciation or depreciation only in the circumstance that it causes the interest rate spread to change direction, which means that a rate hike is not necessarily associated with appreciation or depreciation, but only the degree of appreciation or depreciation as the interest rate spread may increase or decrease under a rate hike.