I don't know. It's very interesting that all three editions share the same pitches in the same order, but the neume representations change, the rhythms are altered, and even quarter-bars are added (and moved).

If I had to guess, I'd propose that
- the 1883 version is closer to the "original," by which I mean the first source(s) consulted. Solesmes notation now includes several rhythmic indicators that they invented, and I've read that the most recent Solesmes editions will be (or perhaps already are) stripping away such markings. So the 1883 edition, without dots or episemas, may be a sort-of "urtext" approach, and perhaps in alignment with the most recent scholarship at Solesmes. (In other words, maybe a 2016 edition would look like the 1883...)
- the alterations in the later versions may have been introduced because Solesmes found additional manuscript sources with which they could compare/flesh out their melodic ideas, and/or they were developing a more comprehensive rhythmic system into which they retrofit the melodies they had already scored.

But to quote Monty Python, "this is my theory, which is mine..." I'm happy for the theory to be corrected by someone who has studied this more assiduously than I.

I think it is because while Pothier was at Solesmes, he was transferred to another abbey in 1893, and his student Dom Moquereau was the one in charge of the 1903 Liber (see quote and linked article below). So a different guy in charge with ~20 years more research, but same abbey name/"brand".

The last version is the same as the 1908 vatican edition (besides the rhythm markings), which was led by Pothier, so it wasn't derived-from/based-on the Moquereau Liber.

Since the vatican edition was now the official version, the Solesmes/Moquereau Liber was changed to match. But since they were allowed to add markings, some of Moquereau's updated/different readings are perpetuated in the modern Liber anyways, for example the salicus marked in the first bar, despite being notated as a scandicus in the vatican edition.

Much confusion exists with regard to whether the Catholic Church officially adopted the “Solesmes Edition” of the chant for the Vatican Edition. To understand the true state of things, one must realize that Dom Joseph Pothier entered the Solesmes congregation in 1859 and published his Liber Gradualis in 1883 along with several other important works on chant. In 1893 Dom Pothier was appointed Prior of Ligugé, a Solesmes daughterhouse. Then, in 1895, he became Abbot of St. Wandrille, another Solesmes daughterhouse. When Dom Pothier left Solesmes, his student Dom Mocquereau took over where he left off, and quickly started experimenting with different methods of publishing chant as well as modifying many of Pothier's ideas. It is enlightening to study these early Mocquereau editions and note the evolution of his notation, sometimes changing from month to month. As stated above, Dom Pothier used his editions as the basis for the Vatican Edition (especially the Graduale and Antiphonale). In this sense, one can say that the Vatican Edition was the “Solesmes chant,” because Pothier published his seminal works while at Solesmes.

-Jeff Ostrowski