I do not know, what Dr Pie will tell about this from philosophical point of wiev (according to his posts he is a philosopher, I presume), but here is what theoretical physicsts mean by "dimensions" and "imbedded dimensions".
By dimensions we mean a degree of freedom to move in.
Therefore, if sheeps (photons, electrons, gravitons, etc.) are only allowed to move in 2 dimensions (=have only 2 degrees of freedom), their universe is 2-dimensional. Therefore, they should obey inverse R law. If sheeps have 3 degrees of freedom, inverse square, 4 degrees - inverse cube and so on.
And vice versa - measuring strength of interaction as a function of distance, you can tell how many degrees of freedom a field (=messenger particles) has. E/magnetism and gravity obeys inverse square, thus photons and gravitons are 3-D objects. Thus we see, eat and fall in 3-D universe.
Having more degrees of freedoms on some length scale (say, on Plank scale) is what we call extra "small" dimensions. Physicists need them to explain all variety of forces (gravity + e/m + nuclear) in simple turms of mecnatical (well, quantum mechanical) motion only. Not seeing any deviation from 3-D on large and small scale, physicists still do not rule out extra degrees of freedom on a very small scale (Plank or so). This is impossible to probe directly (too small), so they rely on mathematical consequences of existence of small degrees of freedom on interactions in "big" 3 dimenstions.
Also, due to gravity electromagnetic interactions in 3 "big" degrees of freedom deviate from interactions at the absence of gravity - you may call it "bent degrees of freedom". So, 2-D sheep universe can be bent, closed, etc - still being 2-D. In this case sometimes it may be mathematically convenient to describe it as 2-D boundary of 3-D body (hyperboloid, sphere, etc) which does NOT mean that sheeps now have 3 degrees of freedom. Theit universe is still 2-dimensional.