For conceptual vocabulary, you have to really learn the kanji. Unlike English, where many synonyms come from different linguistic roots or slight nuances, Japanese words are (usually) based on the kanji. Unfortunately, WaniKani normally just provides the most common meaning for kanji, so sometimes you’ll miss out on secondary meanings if you don’t do extra research.
気 can mean “feeling” with a focus on its meaning as spirit or energy. 気 is used when neither a particular emotion nor physical state can be described stand-alone but rather a general sense. Usually, when 気 is used in a way to denote “feeling,” it indicates some sort of change in the status quo.
感 is “feeling” with a focus towards “sensation.” As you’ll learn later, this kanji is used when referring to the five senses. If you have a “feeling” that someone is watching you, you’re looking at 感.
情 is “feeling” as in emotion. If you have “feelings” for someone or a movie gave you the feels, you’re looking at 情.
気持ち are emotional feelings brought about by a circumstance. If you’re really hot and a cool breeze blows by, it would be 気持ちいい. If you’re shirt starts sticking to you because of the sweat, it would be 気持ち悪い. 気 is your current emotional state, while 持ち describes how it has be “touched.”
感じ most closely resembles “feeling” with a reference to how something “seems,” most appropriately, a “sense/sensation” or “impression.” It’s a very common word that usually describes an abstract concept that invokes a feeling. Say you were about to give a speech. If you describe how your heart was racing quickly, your mind was racing through what you had to say, etc., it’s that sort of sense (感じ).
感情 are “feelings” as in emotions. The combination of physical feelings 感 and emotional feelings 情 are what we know as emotions. 感じ described the effects you experienced, but didn’t describe how you actually felt. 感情 would be that you were very nervous.
気分 is “feeling” as in your mood. If you’re not in the mood to play a game, このゲームを遊ぶ気分になりない is effectively “I can’t get into the mood to play this game.” However, this refers to your mood specifically, and cannot be used to describe someone else’s mood or a general “mood” (such as a two lovers on a date having a good mood going). Those words are covered by other Japanese words. To reference the kanji, 気分 describes that their are many parts 分 of your spirit 気. The current 気 you have is your 気分.
心持ち is “feeling” as in an emotional state. If you can describe someone as a happy personal, you are describing their 心持ち.
To sum up: You wake up right before your alarm feeling refreshed and awake (気分). When you stand up, you stretch, enjoying the feeling of your body loosening up (気持ち). When you walk out your door and see the clear, blue sky with the occasional puffy white cloud, it feels like like a day where nothing can go wrong (感じ). As you walk into work, your boss tells you you’re getting a promotion because you’re a very positive person (心持ち). This makes you incredibly happy (感情).
Hope this helped! If you ever get stumbled by the concepts being synonyms in English, put some research into the kanji to figure out the combo. Not only will this make understanding how to use the word easier, but the time taken to do so will embed the word in your memory.