Does intermittent starting difficulty indicate alternator or starter?

To address your question directly, I'd say neither - I would suspect the alternator. If your alternator is faulty, it would not provide enough 'juice' to power your spark plugs while driving, which would drain the battery and make it more difficult to start the car. A faulty alternator is typically intermittent, which would explain the intermittent issue you've described.

Barring that, and assuming the battery and starter are indeed fine, I would next check the spark plugs and finally fuel injectors.

I don't often see this mentioned but it can be a beast to diagnose because it is intermittent in nature and the problem "internal" to the battery.

Internal to the battery are the "inter-cell" link connectors. (years ago they were external) They can become defective and cause intermittent dead battery symptoms. What occurs is the battery cell connector works itself loose or develops a low impedance path to its adjacent cell. Everything will normally work perfectly and then you may stop for gas and turn off the motor. You then go to try and start the car and it has all the appearance of a completely dead battery. You check the "open cell" battery voltage and it's normal and many times the headlights will work and everything will many times work till you try to start the auto. That's when that small resistance that is internal to the battery becomes significant and the "voltage drop" becomes too great to allow the battery to deliver sufficient starter current.

You may hit the start key several times and magically the car starts and all is normal for an indeterminate time. Jumping may also result in the same phenomenon (all of a sudden all is fine syndrome). What's happening is that the inter-cell connection is literally welding itself together using the high current of jumping or if you can cause a spark when repeatedly hitting the starter.

There is a problem though. Hydrogen in the battery when exposed to a spark can go boom and blow the side off a battery. It has happened twice to me and I now know how to effectively troubleshoot a battery with an inter-cell intermittent. It's not difficult but it requires the ability to use a voltmeter and measuring the "open circuit" voltage (almost always normal) and the voltage at the battery "terminals" with a light vs heavy starting current.

The problem is the battery (because of the intermittent nature of the internal cell connection) can only be accurately troubleshot when the problem appears.

If you run into this type of problem a cheap voltmeter can become your best friend. If you have no starting current, check the "open circuit" (ground cable removed) from the battery and the voltage checked at the battery's terminals. It will almost always be normal. Reconnect the ground cable and re-check the voltage. It will either be completely normal or on occasion the electrical load from the cars computers etc. will cause the voltage to take a significant drop. Then hit the starter while measuring the battery voltage at the terminals. If the battery has an intermittent inter-cell connector problem, the voltage will drop to very close to zero yet if the key is released the voltage will mysteriously go to nearly a full charge condition or whatever it was when the ground cable was reattached as mentioned earlier.

I hope this is helpful. Many pull their hair out trying to figure this problem out as it is intermittent, is not related to temperature, can develop in a nearly new battery and is really a mechanical defect internal to the battery that results in an electrical problem that is more related to vibration than anything else. It can strike at any time and meaningful troubleshooting can only be done when the problem appears... and worst of all seems to repair itself.