I think you are talking about starting a sentence off with a gerund, (a verb ending in ing or ed, acting in place of a noun.) Not to be confused with a participle...
Personally, I see nothing wrong, provided you structure the sentence so the gerund has a possessive subject, and done sparingly In your examples...
Thinking I could save some time, I took the short cut home. This is in first person so the "I" following thinking is the possessive subject that is doing the thinking. If I remember correctly
Now had you said... Thinking to save time, I took the short cut home. while it sounds the same, there is no possessive subject before the comma, so who was thinking?
And even that can be argued and thus a reason gerunds opening a sentence are not favored in most cases to start a sentence, because they are improperly used. Over use, even properly also sort of sounds like a list of actions and thus becomes boring.
Of course, I could be wrong about all of this too, as its been a very long time, but hell, I might open the discussion... lol.