^Oh...uh...When I see a second inversion Bm or whatever inversion of a chord my hand just automatically moves there in the most comfortable position, without any thought as to what fingering I should use.
I've given the 'standard' fingering for basic chords earlier on in this thread. However, it should be noted that the standard fingering can vary, depending on what's coming next, and what has come before.
Each instance is different, so it all depends. Conversely though, if you feel your technique is an issue, then you must feel there is a problem with your fingering?
Learn the basic fingerings for the shapes and try those.
If I trained more on fingering would mean conciously putting an effort to use certain fingering for certain parts of a song, instead of just using whatever is the most comfortable. How would it decrease the time spent for learning a new song?
Because you could learn a piece sitting on the train, or could walk along the sea front with the score in your hand and learn the piece. You could look at a score and see a chord as, for example, a 2nd inv. Bm (as we were saying earlier), and write in above the chord what it is. This way you will know a good chunk of the song (know the chords and the fingering) before having ever played it on your instrument.
Like I said, I personally find Debussy easy to learn this way (PS this is only one of many ways of learning a piece. But the subject is chords, hence focusing on this). I learned Debussy's 12th etude ('pour les accords') this way, and 'knew the notes' before playing it at the piano.
There's a good chance that specific setions you have trouble visualising when purely looking at the score are the same sections you'll have difficulty in actually playing when at the piano. Which makes sense, as your hands only do what your brain tells them to do. So if your mind isn't fully in control of what you are meant to be doing, your fingers never will be.
PS i'm assuming by your user name you're a Mahler fan?!