When selecting your eBird checklist protocol, the phrase 'birding was your primary purpose' is used in several protocol descriptions. This is an exceedingly important aspect of data quality in eBird observations. Our one-sentence definition of this, with stress on the important parts:
"When birding is your primary purpose, you are making an effort to find and record all the birds around you to the best of your ability."
Primary-purpose birding (i.e. complete checklists) might include a traveling count while walking a trail, or a five minute count of birds at a point along the roadside, situations when you are usually outside, on foot, and fully tuned to all birdlife. When birding is not your primary purpose, you might notice a bird of interest and want to enter that single record, but you made no effort to do a more complete survey of the birds around you. Examples might include a Golden Eagle that flew over while you were at your child's football practice, or an incomplete list of birds that you casually noticed throughout the day while gardening. Similarly, a bird flying across the road in front of you or seen while driving at highway speed is by definition not a time when birding was your primary purpose, because you can't reasonably make an effort to find all the birds around you (for example, small landbirds or vocalizing birds would not be detected). The important point is that you did or could not not make a concerted effort to find and record all the birds in these situations.
You can on occasion turn what would be one or more incidental observations into more effort-based checklists, just by the way you note it down. If a Golden Eagle flies over, you could wait around for 5 minutes with the primary purpose of birding, and turn it into a 5-minute stationary count. These counts are better when done unsolicited, as opposed to being prompted by the eagle overhead, but it is still painting a better picture than an Incidental.