How does interest rate impact exchange rate? When I’m using the forward exchange rate formula based on non-arbitrage theory, it tells me that a positive interest rate spread (i_domestic > i_foreign)would cause the forward exchange rate(units of foreign currency that I get for each 1 unit of domestic currency I pay) to be lower than the spot exchange rate, indicating that a positive domestic-to-foreign interest rate spread would lead to depreciation of the domestic currency.

However, people keep saying that if the domestic Central Bank decides to raise the domestic interest rate, it would make the domestic currency to appreciate as international investors would be more willing to pour their money into the domestic market(selling foreign currency for domestic currency) due to a higher yield. This is saying domestic interest has a positive relationship with the exchange rate(units of foreign currency that I get for each 1 unit of domestic currency I pay).

So I’m confused about how these two different approaches give contradictory results. Can someone please explain this to me?
Your first paragraph, except the last part of the sentence, is the statement of covered interest rate parity (CIP), which holds perfectly if not absolutely, except after the 2008 financial crisis (See this paper). However, in the last part of the sentence, if you use forward exchange rate to represent expected future exchange rate, this is where the uncovered interest rate parity comes in, which does not hold according to many papers.

About your second paragraph, the confusion arises from the benchmark that you want to compare. According to CIP, if the domestic interest rate is higher than foreign interest rate, then the spot exchange rate is less than the forward exchange rate (in domestic currency per foreign currency). In other words, if you use forward rate to represent expected future exchange rate, you expect that the domestic currency will depreciate and the foreign currency will appreciate. In this case, you are comparing spot rate and forward rate at the same point in time.

If, suddenly and unexpectedly, the domestic central bank raises interest rate, immediately you will observe a sharp drop in the spot rate (in domestic currency per foreign currency). So if you compare the spot rates before and after raising the interest rate, you find that the domestic currency appreciate. However, whether you stand at the time before or after raising the interest rate, you always observe that the spot rate is less than the forward rate (in domestic currency per foreign currency) as long as the domestic interest rate is greater than the foreign interest rate.