What learning materials are you using? (Textbook, website?) Your source doesn't seem very good. There are a lot more uses of the te-form, and by itself it can't mean "presently happening/doing". You need to add iru (or imasu) for that.
Don't worry about trying to do something like learn "all" ways to use the te-form at once, though, because that would be a bit overwhelming. Learn them as you go, and make sure you understand how to use each properly. For example, do you know what the difference is between just te form, and te form + kudasai, in asking someone to do something?