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I know the total current inside a BJT has to be constant all throughout the device, but the figures above seem to suggest that the total current in the emitter should be larger than the total current in the collector. Since we have extra hole current in the emitter going the opposite direction as the electron current (the conventional current for both have the same direction), shouldn't it be that the current in the emitter region is higher than the current in the collector region?
The current in the emitter region is indeed larger than in the collector, because the emitter current has to be equal to the collector current plus the base current. That's just KCL.
Just like \$β\$ is defined as \$\frac{I_c}{I_b}\$, you can also define \$α=\frac{I_c}{I_e}\$, which is always less than one--i.e., \$I_c<I_e\$. (α is also often seen defined as \$\frac{β}{β+1}\$, which can be derived from the definitions given above and KCL.)
You are correct, emitter current is base current + collector current. In fact, emitter current is I(emitter) = I(base) + I(collector)
And you can calculate Beta from measured values... Beta = I(collector) / I(base)
Then calculate exactly how much current is in each part of the transistor.
