The simulation result is correct. What you are seeing is Ibc: Current flowing from base to collector. Remember that the base collector junction is forward-biased.
Normally we don't think about the current flowing in that direction. But it can/does.
In saturation, some base current flows to the collector. The net flow may be from outside of the transistor into the collector, but that does not mean that Ibc = 0. This is something I did not fully grasp until quite recently, and I even down-voted someone over it. I mean, I am sure I knew it at one time when I was in school. But I must have forgotten it long ago.
Given that \$\alpha\$ and \$\beta\$ are related by \$\alpha = \frac{\beta}{1+\beta}\$ as stated in the wiki article, obviously you can do your sums with either.
However, which is going to be easier to use? I personally always use \$\beta\$, regardless of the transistor configuration.
In common emitter \$I_c = \beta\times I_b\$, so I can say 'I need to control \$I_c\$ collector current, I need at least \$\frac{I_c}{\beta}\$ of base current to do it'.
But as \$\beta >> 1\$ (for most transistors), \$\alpha \approx 1\$, and \$I_c \approx I_e\$. You may object to the approximation, but given the way that \$\beta\$ varies with temperature, \$I_c\$, and between transistors of the same type, that is a far far better approximation than insisting that \$\beta\$ is constant. Any good transistor design will allow for operation with a range of \$\beta\$, at least \$2:1\$, preferably more.
Once you have made the approximation \$I_c \approx I_e\$, then common collector operation is given by 'I need to allow for a base current of \$\frac{I_c}{\beta}\$ to flow in the base circuit, without upsetting operation'.
With a common base stage, you say much the same thing, allowing an amount of base current, however you also say that the emitter to collector gain is slightly less than \$1\$, a fraction of \$\frac{1}{\beta}\$ less than one. The error of the gain from \$1\$ will usually be a smaller error than resistor tolerances and other sources of gain error.
Given that you can write an equation for \$\alpha\$, does that mean that you need to? For most practical engineering designs, the answer is no. If you are in college, and the tutor really likes to use \$\alpha\$, then the answer is yes.
Best Answer
At first, you have to realize that the emitter current - and accordingly the collector current - is controlled by the base-emitter voltage Vbe (see Shockleys famous exponential equation). I am aware that in some textbooks the BJT is described as a current-controlled device, but that is a simplified description which cannot explain all observable effects. The base current does exist (unfortunately) and it is taken into account for calculating the base bias circuitry - but it is kind of by product. (Barrie Gilbert: ...the base current is purely incidental - it is best viewed as a „defect“).
Now - what happens if the voltage Vce across the BJT increases? This increase results in a broader depletion layer between collector and base (Vcb increases). As a consequence, the space charge region between base and emitter (forward biased by Vbe) decreases. Therefore, the electrical field strength within this area is increased (constant base-emitter voltage) - and more charged carriers are able to move to the collector. This effect somewhat increases the collector current (and reduces the base current for a constant emitter current because of Vbe=const.).
Note: This happens even in the case that the base current is kept constant (Ib=const.). Hence, this is one of some proofs that it is the voltage Vbe which controls Ic.