Electricity actually flows from minus to plus: in metals, electricity is carried by electrons and electrons are negatively charged. The idea that current flows from positive to negative is a historical mistake made before it was realized the current carrier actually has a negative charge.
So, by your reasoning, slightly corrected, you should connect only the minus. That's untrue.
The problem here is that a battery is a charge pump. If the electric potential between its terminals is less than about 13 volts, it takes an electron from its positive terminal and moves it to its negative terminal (actually they are different electrons to be precise, but the effect is the same, electron disappears from one terminal and appears at another terminal). This forces the potential difference to be always about 13 volts (ignoring internal resistance here).
Let's assume the failed car battery has exactly zero volts, for simplicity.
Let's also assume the failed car battery has a potential difference with respect to Earth of plus 3000 volts (very typical, cars are notorious for creating static electricity and static electricity has a surprisingly high voltage). Both negative and positive terminal is at 3000 volts.
Let's now assume the good car battery negative terminal has a potential difference with respect to Earth of plus 1500 volts. This means the good car battery positive terminal has a potential difference with respect to Earth of plus 1513 volts. The battery as a charge pump maintains the difference of 1513-1500 = 13 volts.
Now if you connect the negative cable only, it will rapidly equalize the negative terminals. If we assume the grounded parts of both cars to have same charge storage capacity, they will cause the connected-together negative terminals to equalize to the average of their original voltages, which is 2250 volts. This happens by electrons flowing from the good car battery negative terminal to the bad car battery negative terminal. You may see a very small spark of static electricity during the connection, but you'd have to be in a very dark environment and have an excellent eyesight to see the minor spark of static electricity.
The good car battery is a device that ensures the potential difference between its terminals is always 13 volts. So the charge equalization between negative terminals ramps up the voltage difference of negative terminal with respect to Earth. This threatens to cause the potential difference between its terminals to fall below 13 volts. The battery as a charge pump thus takes electrons from the positive terminal and moves them to the negative terminal. This causes the positive terminal to move to a higher electric potential.
So now you have this:
- Good car battery negative terminal: 2250 volts
- Good car battery positive terminal: 2263 volts
- Bad car battery negative terminal: 2250 volts
- Bad car battery positive terminal: 2250 volts
There will be no charges moving after that. You haven't completed a circuit by only connecting the negative lead.
Now what happens if you connect the positive too? Because the good car battery positive terminal is at 2263 volts, higher than bad car battery positive terminal at 2250 volts, electrons want to start moving to a higher potential, because they are attracted by the positive charge of the higher potential. So electrons in the bad car battery positive terminal start moving towards the good car battery positive terminal. This gradually raises the potential of the bad car battery positive terminal, to 2263 volts in fact, because less electrons means more positive. The electrons moving towards the good car battery positive terminal lower its potential, so the potential difference between the good car battery positive and negative terminals threatens to fall below 13 volts. The good battery is a charge pump, so it moves electrons to the negative terminal, maintaining the 13 volts of voltage difference.
What you then have is the following equilibrium:
- Good car battery negative terminal: 2250 volts
- Good car battery positive terminal: 2263 volts
- Bad car battery negative terminal: 2250 volts
- Bad car battery positive terminal: 2263 volts
Now there are two possibilities. Either the bad car battery is completely dead. This means the 13 volts of voltage difference between its terminals no longer causes any charging current. If this happens, the situation stays in the equilibrium. However, it's also possible the continuous 13 volts of voltage difference between the bad car battery terminals causes a continuous charging current to flow (in which case to be precise the potential difference at different points will be different due to Ohm's law, resistance of metals, and resistance of batteries, but if we assume most internal resistance is at the bad battery, then the same voltage difference of 13 volts is a good approximation). If this means, you have the following circuit with continuous current:
- Electron at good car battery positive terminal moves to good car battery negative terminal (actually this is not the same electron as it causes chemical reactions to happen, the chemical reactions slowly deplete the battery state of charge)
- Electron moves from good car battery negative terminal to bad car battery negative terminal
- Electron moves from bad car battery negative terminal to bad car battery positive terminal (actually this is not the same electron as it causes reverse chemical reactions to happen, the reverse chemical reactions slowly increase the battery state of charge)
- Electron moves from bad car battery positive terminal to good car battery positive terminal
This continues as long as the state of charge in the batteries is different. Once they have the same state of charge, the process stops.