During Update 9, I found the following (with assistance from Lithic's data):

Code:
Level 	 Lessers
1	 1d3
2	 2d3
3	 4d3
4	 6d3 + 4
5	 10d3 + 12
And for greaters, a 1 in 3 chance of generating any number up to and including the level of deconstruction. Hence, the average return for each level of deconstruction were:

Code:
Level 	Lessers	Greaters
1	2	1/3
2	4	1/2
3	8	2/3
4	16	5/6
5	32	1
Two changes with Update 9.1 stated:
"Deconstructing low tier effects on high-end loot will now return more reward essences."
"It is now possible to deconstruct crafting Shards using the Deconstruction Eldritch Device."

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First, let us investigate what it means to be "high-end" and "loot". I have observed several divergences from the patterns found above. The format I will be using is "item's minimum level - deconstruction level : lesser essences, greater essences". Thus, "8 - 1: 4, 1" refers to a minimum level 8 item, level 1 deconstruction, which returned 4 lesser essences and 1 greater essence. (An "r" following the item's minimum level denotes that it was race-restricted.)

8 - 1: 4, 1
6 - 2: 12, 2
10 - 2: 7, 0
10 - 2: 7, 0
12 - 2: 3, 3
12 - 2: 4, 3
12 - 2: 4, 3
10r - 2: 6, 3
10 - 3: 35, 1
10 - 3: 38, 0
13 - 3: 14, 2
14 - 3: 13, 0
14 - 3: 13, 2
10 - 4: 24, 5
12 - 4: 24, 0
14 - 4: 37, 0
15 - 4: 21, 5
16 - 4: 23, 1
16 - 4: 24, 4
11 - 5: 36, 6
12 - 5: 40, 6
12 - 5: 46, 0
12 - 5: 48, 1
15 - 5: 43, 0
16 - 5: 33, 6
13r - 5: 33, 6

All told, 15 instances of more lessers than expected, 9 instances of more greaters than expected, and 2 instances of both more lessers and more greaters. The overall rate of aberration was 7.5% +/- 10.8%. As the level of error indicates, comparing rate of aberration on a decon level basis (or even worse, a minimum level basis) is useless at this point.

In any event, we can notice immediately that all levels of deconstruction have a chance of producing unusually high rewards. We can also notice that what it means to be "high-end" is very oddly defined - a mere minimum level 6 item produced unusually high rewards. It is also interesting that the aberrant greaters are never more than +1 from the original distribution.

It is worth noting the possibility that the underlying distributions were also changed, that perhaps a level 1 deconstruction is now a 1d4 rather than a 1d3. This is unlikely: over 41 level 1 deconstructions, I have 11 1s, 18 2s, 11 3s, and exactly 1 4, which came on one of two minimum level 8 item level 1 deconstructions.

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Next, I will treat Shards vs. Items. Both the language of the release notes and my findings suggest that Shards do not receive the "more reward essences", which is to say they are not considered "loot", although it's conceivable that this is only because the shards I have deconstructed so far aren't "high-end" enough yet. How the equivalent of item minimum level is determined on a shard is anyone's guess, but for now I have lots of data on +1 and +2 shards and have seen nothing out of the ordinary. For a +2 shard level 2 deconstruction, I have:

Code:
Lessers	Count
2	12
3	40
4	55
5	37
6	21
This fits a 2d3 distribution nicely. By comparison, my non-shard level 2 deconstructions have:

Code:
Lessers	Count
2	8
3	14
4	32
5	27
6	12
7	2
8	0
9	0
10	0
11	0
12	1
.

Now, there is also the question of the chance of generating greaters at all. Before update 9.1, I hypothesized that this was simply d3 across all deconstruction levels, and the data was consistent with that hypothesis. As mentioned earlier, I do not have enough data yet to make a worthwhile comparison between minimum levels, or between deconstruction levels, but I do have enough to look at between Shards and Items. Currently, the rates I have are 34.5% +/- 13.0% vs. 36.7% +/- 10.8%, respectively. My guess is these will converge on 33%, and therefore that "more reward essences" refers specifically to the number generated rather than the chance of generation (in the case of greaters). In any event, there is no evidence currently to suggest that Shards have less of a chance of generating greater essences than Items.