I accidentally started up my engine after draining the oil and inserting the oil filter. I know that running an engine without oil causes the parts not to be lubricated and thus, causing damage. In my scenario, if the oil change was an hour before it was run on empty is the likelihood of damage high? I’m interested to know because aren’t the parts still lubricated from the old oil? It was only run for a few seconds before being quickly turned off.
After you change the oil and filter and refill the oil pan, you have a few seconds before the oil fills the filter and oil passages and begins lubricating the engine again. During that brief period you're relying on the residual oil film that clings to the various surfaces in the engine to provide protection.
How long does that protection last? Generally only a few seconds and after that wear starts accumulating. But it's usually heat build up due to lack of lubrication that causes an engine to destroy itself in the no oil situation.
In your case it's likely there was no more damage than a typical startup after an oil change and if the engine was quickly shut off then chances are good the engine will still live a normal life.
There is little that makes sense to do at this point anyway as any solution is going to be exceeding expensive. If there is damage, either a total rebuild or a replacement engine is called for. This doesn't make sense to do just in case. If you damaged your engine you will know it in time and you can deal with it then.
In the future, take precautions to prevent yourself from doing this. I like to remove the keys from the ignition and put them with the new oil containers. That way I can see that the oil is there and not in the engine if the urge to start the engine comes to my mind.
Some damage is likely, but not quantifiable.
Some engines have been run for 10 minutes because they wanted to destroy the engine and it took longer.
You probably reduced the life span of the engine or it might make a different noise soon.
If it is now filled with oil and running then use it, not much else to do. You could strip it and rebuld it but as it is not clanging knocking then it could just be an expensive exercise for little gain.