It looks like the confusion was caused by bad website design, having a button "show/hide inversions" which shows inversions AND VOICINGS which is a separate concept. But I'll add this anyway, maybe it helps someone. If it doesn't help, feel free to ignore and/or downvote.
The best way to learn a culture and its language is to get involved in the culture, interact with people and do things that people do in that culture. You don't really learn to operate in a culture by implementing descriptions of it in a computer program. Writing music programs will, I hope, lead to discovering different musical practices and incompatibilities between theories and practices. But that's only my opinion.
That being cleared, some more points:
- Chord symbols have evolved primarily to serve as instructions for accompanists in popular music. Short descriptions of the essential harmonic changes in a song arrangement, written for a guitarist, pianist, bass player to play, to improvise accompaniment. The purpose of this is commercial, industrial, hands-on practical, to get things done efficiently with minimal effort.
- There is some kind of a theoretical model behind chord symbol notation: tertian harmony where harmonic ideas can be expressed as stacks of thirds. If something cannot be expressed as a stack of thirds, then it's kind of a non-chord from that perspective. But that's an approximation with limited usefulness in practice. There are many harmonic constructs and aspects that are used in practice, but which don't fit the stack-of-thirds model and either have to be kludged-around or simply cannot be expressed at all. Exceptions have been made to serve practical needs. For example the "6" chord, which according to the pure tertian model is an inversion of a minor seventh chord. Or the "5" chord which means a power chord. These haven't been decided and approved in any standards committee, people just start calling something by some name and sometimes the practice catches on.
- A theoretical "11" chord, if we just systematically follow the idea of adding more thirds to the stack, should contain six different notes, a stack of five thirds. It's a word that can be constructed with the syntactic rules of the theoretical chord symbol language, like you could synthetically construct words like "undescribabilitificationness". Is that syntactically valid? Even if it is, I don't know what it would mean in practice, and nobody uses that word.
- In practice, the "11" chord type has been used to describe a kind of a sus chord, one that doesn't have a third. Why? Maybe just because this chord naming language was the only one that was available. It was a name that was possible to say in that language, so someone thought, it's a word that's free to be used for something so I'll start using it for this chord that I actually use, and no musician in their right mind will try to play a third in there because it would be nonsensical. And now in theory books, there have to be hand-waving explanations that "the third is usually omitted." But isn't that equally ridiculous as explaining that "in practice, there is usually no ham in a hamburger, even though the word ham can be found in the beginning of the word hamburger?" The truth is that you cannot systematically deduce the word's real-world meaning without knowing the culture.
Then the questions
How do I label the chord for the bottom image?
Depending on what you want to achieve and who the players are for who you're writing this notation, you could use at least the following labels in a song chart:
- A11 (rely on your players knowing about musical practice)
- G6/A (gets all the notes right, but if a guitarist plays a full barre chord, the low G will clash with a bass player and sound muddy)
- G/A (gets the sound mostly right, and not too confusing, but the same problem with beginner guitarists clashing with bass)
- A7 sus4 add9 (clunky but if you want all the same notes without a slash)
- A7 sus4 (might be doable for many beginners, only the 9th is missing and no wrong notes added, and no bass clash)
- Asus (even more doable for beginners, gets the "sus" thing right, and no wrong notes)
I slightly question the idea of playing thick piano chords like this on one hand, if played with the exact voicing shown in the picture. Depending on the octave it may sound muddy.
If playing, say, an arranger/accompaniment keyboard with left-hand chords, the keyboard may not correctly identify the A-B-D-E-G chord shown in the picture and it may be necessary to leave out the B, and then Asus4 or A7sus4 would work better.
Is it, or is it not an inversion?
It's not an inversion. Except if you insist on seeing it as G69/A, then it would be an inversion.
Furthermore, if taken according to a pure theoretical definition, it cannot be said to be an inversion of ANY stack of thirds at all.
I entered the notes into the Python program here: Formal definition of an inverted chord
and thus printed the computer:
['A', 'B', 'D', 'E', 'G'] cannot be said to be an inversion of any canonical stack of thirds.
