I agree with you that the description of the reduction procedure on the Wikipedia page is inconsistent with the reduction procedure used in the picture. The curious thing is that different sources seem to have a different idea of what a Wallace tree exactly is. There seems to be no rigorous description of the Wallace reduction phase.
During lecture, I learned to reduce Wallace trees the way it is explained on Wikipedia: the 'greedy' way (reduce as much as you can). But I also found this site, where they use a similar reduction scheme as in the image in your question (in fact, it is meant to be the same, but the Wikipedia image contains an error, in the 4th dot diagram, the 7th row from the right contains a dot too many), and refer to it as 'traditional Wallace reduction'. So there are at least two reduction schemes that have Wallace's name attached to it (and the many papers which refer to Wallace trees or Wallace multipliers in some way suggest that there are many more).
In such cases of confusion, it is often useful to look up the original paper, as it usually helps to understand the historical context of the term. It is not very hard to read, and is, at some points, less formal than one might expect (which might be a source of confusion).
When reading the paper, it becomes clear that Wallace trees were not designed with dot diagrams in mind. Once one has adapted this concept, it becomes very easy to improve the reduction proposed by Christopher Wallace. My guess is that people just used the term 'Wallace tree' to refer to these improved schemes as well. The wealth of papers and distinct designs that one find when googling for 'Wallace tree multiplier' seems to indicate that the term is used quite loosely indeed.
Now, let's take a look at the dot diagram of the Wallace tree in your question. Compared to a Dadda tree, two things stand out:
- The tree is not shaped as a triangle, but rather as a diamond or parallelogram
- There seems to be some division of rows in groups of three, and adders do not cross boundaries. After a reduction step, the dots stay within groups of two rows.
To understand this, it is useful to note that in the original paper, Christopher Wallace does not use dot diagrams. Instead, he loosely describes the strategy as "group three numbers and reduce them to two using adders". The way he builds trees is to take a number of bits with the same weight, and build a tree of those using only full adders. This is illustrated in figure 1 from the paper, in which 39 operands are added: to add 39 n-bit numbers, you would need n of these trees (plus some structure to handle the carry bits, which could be done with simplified versions of these tree, as it involves adding less than 39 bits: this is where dot diagrams come in handy).
So, Chris Wallace was not thinking about dot diagrams when he came up with his method (which is a reason that it is a pretty inefficient way of reducing numbers in some aspects). He was thinking in terms of numbers (more specifically, multi-bit numbers).
When you think about it, this explains the layout of the dot diagram. The groupings of rows are groups of three numbers which are added together. From this perspective, it doesn't make to much sense to mix bits of the numbers. This 'traditional' Wallace reduction method is really quite trivial when you think of the rows of dots as multi-bit numbers, and you just reduce each group of three numbers to two numbers.