Dear emaycom,
Hope you're well. On the couple of occasions when an alternator belt failed on my cars, all - not just some - of the warning lights came on. I'd guess Volvo did this, to grab the operator's attention. It seems to work.
Some will ignore a single warning light, especially if the car's performance seems fine. That's the case here: your car ran smoothly "on battery only", after the broken belt stopped the alternator from re-charging the battery.
Depending on the battery's condition, a car may run on battery only for an hour or so. When the battery gives its last, the car stops. Recharging a fully-discharged battery is pretty time consuming. If the battery is weak, the full discharge may "kill" the battery: it won't recharge.
I doubt the brief period, when the battery was not re-charged, did any damage. Replace the alternator belt and all should be well. Check the other drive belts: it is best to replace drive belts in sets.
If the belts have small cracks in them, they don't owe you a nickle. It is better to replace them in your garage/driveway, than road-side.
Hope this helps.
Yours faithfully,
Spook
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