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I use one of the Android phone apps for ear training, and one of the drills is to recognise chord inversions, this includes minor/major chords and also dominant 7-th chords. They conventionally label inversions with numbers from 0 to 3 with 0 being the root position.

I trained myself to recognise the harmonic versions of the chords by listening to the top note, so that the recognition of the inversion becomes equivalent to recognition of a kind of melody.

One of the advanced exercises is recognising inversions of the open chords, and they contain sometimes 5-6 notes. My strategy with the highest note failed because it seems that this is not a proper definition of the chord inversion.

Question 1. Conventionally speaking, what does it mean - an inversion number of the chord in open position? Does such a convention exist?

I suspect that there is either no convention, or the program assumes the position of the root note, but I don't know any reference.

Question 2. Is there any practical advice to recognition of the inversions of the chords, especially in open positions?

Currently my way is just a "dumb practice", i.e. trying to listen and recognise. But I suspect that there may exist more useful exercises designed especially to learn how the chord is presented.

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Inversions are always worked out from the bottom note. Thus using a C triad, in closed form, C E G is root, E G C is 1st inversion, and G C E is second. That's equivalent to your app's root=0, 1=1st inv., 2=2nd inv. and 3=3rd inv.

In open form, C at the bottom makes it a root, E at the bottom makes it 1st inversion, and G makes it second. That is regardless of the other two notes: C E G and C G E both get called root position.

When encountering the seventh notes (usually) the same rule applies, but obviously in closed position there are root, 1st, 2nd and 3rd inversions with only one order, bottom to top, but in open voicings there are many more. However the same base note ruling applies.

So, you trying to ascertain inversions using the top note as a datum point isn't always going to work. And, yes, there may be times when certain notes are doubled - take a guitar playing a four note chord - at least two of those notes will be doubled. We talk about drop chords - which may be of interest - at this point.

Q2. Simple solution for answers - listen to the lowest note!

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    In the first paragraph, you say "equivelant to... 3=3rd inv." but your previous sentence only spelled triad inversions. Might be worth explicitly noting that the third inversion would be for a four note chord, the seventh being in the bottom. – Aethenosity Jul 27 '18 at 13:23

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