You wrote, "That is about object permanance. It isn't about measurement."
I don't believe you can have object permanance (in your mind) without having omitted measurements. To begin with new borns don't even have a sense of objects - not as unique or discrete entities. I refer you to James' description of the baby's world of buzzing, blooming sensations - just a blur of colors and sounds that baby hasn't learned to percieve as separate entitites.
Without entities it isn't possible to have a clear understanding - even on the simplist level - of events, attributes, actions, causes, nada!
The heart of this disagreement seems to be in what is meant by "omitting measurements."
You wrote, "Why does a concept of a thing entail measurement? If a child sees some clouds and develops a rudimentary concept of cloud, is it necessary for the child to measure the clouds to have the concept? In the case of the lion, the issue is object permanance. The baby is not measuring anything, nor (I presume) even knows what measurement is or how to use a measuring instrument, e.g. a tape measure.
'Omit' has a dual meaning. One can be aware of something and then ignore it. One can omit something by not being aware of it all. Is the latter what "implicit measurement" means? If yes, then it must be cognitively irrelevant."
A lot of the newer research done with babies in this area uses expectations. Because babies don't have the ability to communicate the researchers are attempting to determine what the baby expects and what constitutes a violation of the expectations. And then they make judgements of what that tells them about what the babies cognitive development would have to be to have those expectations. If a baby made an error in what was omitted then reality may present an event that violates the baby's idea of what a given thing can do. Baby will have to re-integrate based upon omitting a better measurement (e.g, Dog = fun furry thing.... until baby is bitten at which point it drops the fun part.)
It has been likened to magic. You are watching some sleight of hand being done and a coin appears to disappear. You expect object permanence, and object visibility, and thus you are surprised when these expectations appear to have been violated. This surprise is a form of psychic dissonance that the mind wants to resolve. Baby's learn the properties of things and integerate new knowledge with old by focusing on this kind of surprise.
When a baby watches clouds they can be learning to identify them as objects. That is an act of volitional focus. The senses bring in undifferentiated sense data. Some part of the mind initiates focus with the implicit question of "is that a thing?" From that instance it is a percept and will automatically be seen as such. Then they can decide that differences in shapes and shades of color are omitted. The only measurement that is happening here is the noticing that thing A has a shape, and that thing B has a shape. And even that focus may not happen except as part of the realization that both A and B are similar things. That is the 'aha' moment even though the measurement omitted was implicit.
Now, here is where I think the heart of the disagreement exists. Nearly all of our mental heavy lifting is done by the subconcious. It is far more powerful and extraordinarly good at symbolic work. I believe that the "implicit measurement omission" is where there is no question that a measurement was omitted, yet it wasn't done explicitly, and the person hasn't become explicitly aware of what allows them to say, "Yes, that is a cloud, and that, and that, and that, and that, but no, not this thing over here" and be accurate. They would need to ask themselves, "How do I know that, and study the concretes to come up with what they have omitted."
Notice that "measurement" is used in its generic form. It does not mean creating a math formula or a taxonomic set of descriptors that allows cataloging of clouds by individual shapes such that one could quantitatively distinguish between two similarly shaped clouds. The meteorologist can come back and add to our knowledge of the specific measurements, e.g., Cumulonimbus is a type of cloud from the family of vertically developed clouds, that is 6,500 to 60,000 feet tall, dense, and involved in thunderstorms and other intense weather.
Our individual capacity to integerate and to reason on principles will depend upon how much of our measurement omission we have made conscious. Most scientific and engineering work comes out of an explicit study of principles that are explicit derivations of what would have started as implicit. And it isn't a simple implicit versus explict. There are degrees of consciousness.
After I discussed the levels of development (concieve, symbolize, consistent symbology, express, express in common language), you asked how this related to measurement. I wasn't intending it to be just about measurement, but about the development process in general and how it was tied to communications. But because abtraction and integration are so basic, and measurement omission is so basic, they are tied into all of our processes. The baby is having to omit measurements to grasp concepts in many areas - including coming up with the idea of communicating.