I think that the bath tub analogy actually works like this (since you cannot physically increase the size of the bath tub but you can shuffle the water around):
bathtub = stock
amount of water = float
water level = share price
If fat guy wants in without raising the water level, he must siphon some water (equal to the volume of his body) out of the tub into a bucket (= lending pool that shorts can draw from). Once he's in, he needs the water so he siphons the water back into the tub but the tub water level (= share price) must rise in proportion to the volume of his body. But the lending pool is empty.
NB: No wonder IB reports today no more shares to lend since the back siphoning has begun. Watch the water level rise now in the tub. (it took me a while to figure out how that analogy works last night during sleep hehe...)