How does interest rate impact exchange rate?
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.