It's about categorizing the language into "Objective or Subjective (not she, her ..)" depending on the meaning.
1. Objective meaning : do (perfect tense included), auxiliary verbs (not subjunctive), relative-clause, conjunction-clause.
2. Subjective meaning (In speaker's mind, personal, emotional) : to do (yet happened) , ~ing (now), ~ed (happened).
For example ...
 do Vs. ~ing; -ed
- The book sells well. (The fact that many customers buy this book. objective)
- The book is selling well. (From the speaker's judging and opinion and thought in his mind, many customers buy this book now, subjective)
- The book is well sold. (From the speaker's judging and opinion and thought in his mind, many customers bought this book. subjective)
- He has gone. (Indicates the fact that he left. obj)
- He is going. (In his mind, he is going now. subj)
- He is gone. (Judging from his thought, he left. subj)
- I'm dying to drink water. (In the speaker's mind, emotional meaning)
cf) I die to drink water. (literally die, obj)
- He's being kind. / She's always crying. / I'm lovin' it! (Based on the speaker's thought, emotion)
cf) He's kind. / She always cries. (obj. but you can't cry always) / I love it.
- I invited you. (Fact)
- You're invited. (So please come - additional emotional meaning. subj)
 auxiliary verbs Vs. to do
- I will kill you. (I plan to kill you. obj)
- I'm to kill you. (I want to kill you. I could, would, might, should kill you. ... subj)
- If I'm to be killed a hundred times, ~ (In the speaker's mind. subj)
cf) If I'll be killed a hundred times, ~ (obj. but you can't be killed a hundred times.)
 phrase (to do, ~ing, ~ed) Vs. clause (relative, conjunction)
- I'm glad that I meet you. (I'm glad for the fact that I meet you. obj)
- I'm glad to meet you. (I'm glad for the personal reason. subj)
- If you do so, you would feel better. (if-clause, from your point of view, objective)
- Doing so, you would feel better. (From my point of view = personal, subjective)
- Birds singing in the morning, would sing for hunger. (In the speaker's mind, hypothetical birds, subj)
- Birds that sing in the morning, (fact, obj.)
Thank you for reading a little bit long post. I wonder if you've heard this before and we can generalize this?
I have not heard it before. Some of the verb forms that you have used as examples seem to follow the subjective/objective dichotomy, but others do not at all, or apply in only one of their uses. I think it is too simple-minded to be more than a vague guideline in thinking about the varieties of verb forms.
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whatever is subjective is not always objective
this distinction is itself not absolute
People are waiting to help.
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