I think you are correct for this hypothetical 2D being. However, you should be aware that time is not just a dimension. In space-dimensions, you can in principle move freely forward and backward, while in time, your motion is fixed.
With respect to the brain scan: this way of visualisation is chosen for simplicity. A regular 3D image, where you can look at any depth you want, will give a clearer image of what is happening in this third dimension. Some information is a bit lost for the observer: you clearly see structures in the x,y-plane, but for vertical coordinate, it is not that obvious.
Some bit off-topic reading material may be: http://en.wikipedia.org/wiki/Flatland
In this context a spatial "dimension" is not the same as the dimensions of momentum used in dimensional analysis. As you point out yourself, you can observe three dimensions - but you can't observe the dimensions of momentum.
The easiest way to think about these spatial dimensions is to look at it from the point of view of relativity. Ever since special relativity, we've had this equation that puts time and space on an equal footing:
$ds^2 = -dt^2 + dx^2 + dy^2 + dz^2$
This is the so-called Minkowski metric. Time ($t$) is different from space ($x, y, z$) because they differ by a minus sign.
The form of this equation obviously suggests a way to extend it to more dimensions. For example say you discover a new dimension, then we simply have:
$ds^2 = -dt^2 + dx^2 + dy^2 + dz^2 \pm da^2$
If your new dimension is a temporal one, then it takes the minus sign, and if it's a spatial one, it takes the positive sign.
This explanation is pretty simplified (the Minkowski metric applies only in empty space for example) but the idea is there.
Best Answer
You say:
but this is not so. We experience three (spatial) dimensions, but there is no distinction between the first, second and third dimensions. For example I might choose the first, $x$, and second, $y$, dimensions to be horizontal and the third $z$, dimension to be vertical. I live in the UK, but suppose a friend in the US does the same, our dimensions would be different:
My first, $x$, dimension isn't the same as my friend's $x$ dimension. So which of us is correct? Well, neither of us. There isn't a unique first dimension; we can arrange our dimensions at whatever angles we want - it doesn't make sense to talk about a first, second and third dimension because the distinction is a matter of choice. All we can be sure of is that there will be three dimensions.
We can image a plane cutting through our three dimensional space, and this plane would be a two dimensional object. However our 2D plane is a purely mathematical construct and no 2D objects really exist.
If instead of a plane I draw a line, then the line is a 1D object. However, like the 2D plane, this is just a mathematical construct.
The point I've attempted to make above is that the universe and everything in it is three dimensional, so it doesn't make physical sense to talk about matter constrained to lie in one dimension.