Why is it that we start to see more stars (than we did at first glance) after staring at the night sky for some time? I realised that if I fix my gaze on a random spot in the night sky, I start to see more stars (those presumably dimmer ones) appearing, if I look long enough. (Note: there is in the presence of dim external city lights.)
Is this due to our eyes adjusting to the dimness of the night sky (can be explained with biology) (first case) or could it be due to a slower registering of the light from dimmer stars, which were of lower intensity and hence at first glance was not detected and recognised as distinct light sources/stars (second case)? 
I do not think that it is the first case - which can be explained (I suppose?) with the same reasons to why we get momentarily "blinded" when walking into a dark room from a brighter surrounding (reason: not enough rhodopsin, the chemical that helps us perceive light in the dark). I disagree because  I was already out in the dark in most occasions and my eyes would have already adjusted to the surrounding brightness(dimness). Hence in that case, wouldn't my eyes be sensitive enough that I will be able to see the dimmer stars immediately upon looking into the night sky/at first glance which I don't?
But since that is not the case according to my experience, I am thinking if it could be the second case? An analogy I thought of (more to help express myself) for this case is by likening our eyes to a blank canvas and the light from the stars are - instead of light waves - single straight streams of (separate) ink droplets, with greater distance between droplets from dimmer stars (lower intensity light) and shorter distance between droplets from brighter stars (higher intensity light). So the longer I look at the sky, the more droplets collect on my canvas, and the lower intensity light would take longer to reach the same intensity as higher intensity light (e.g. in 1s, 5 drops collect on the point corresponding to high intensity lightand only 1 drop on the spot of lower intensity light). Assuming the droplets get absorbed the moment it touches my canvas and only at that spot of contact, it will take longer for our eyes to register that there is indeed a (dimmer) star at x location. In that case, it is not really of our eyes adjusting to the surrounding brightness, but instead of this sudden awareness when we realise over time the difference in light intensity from certain dimmer star spots - aka the moment I realise there are actually more stars than the one I saw at first glance. 
I would like to know the correct scientific explanation (first/second case or none) behind this phenomena of seeing more stars after longer exposure to the night sky, and whether my analogy is flawed?
 A: I don't mean to offend you, but in my opinion  you are over thinking the issue.
From a physics perspective:
The apparent brightness of the stars is measured using the system described below. To sum it up: A magnitude 5 star is 2.5 times the apparent  brightness of a magnitude 6 star, that is  one of the reasons your eyes take a long time to adjust to see the dimmest, (magnitude 6 stars), that are, for most people, the limit of their visual acuity.

Stars of various brightness,  and when your eyes fully adapt, you may see the Milky Way, if the sky is dark enough.

The brighter an object appears, the lower the value of its magnitude, with the brightest objects reaching negative values. The Sun has an apparent magnitude of −27, the full moon −13, the brightest planet Venus measures −5, and Sirius, the brightest visible star in the night sky, is at −1.5. An apparent magnitude can also be assigned to man-made objects in Earth orbit. The brightest satellite flares are ranked at −9, and the International Space Station appears at a magnitude of −6. Since the scale is logarithmic, each step of one magnitude changes the brightness by a factor of about 2.512. A magnitude 1 star is exactly a hundred times brighter than a magnitude 6 star, as the difference of five magnitude steps corresponds to (2.512)5 or 100.

From a biology perspective:
As Jon says above, your eyes simply adjust to the dark. Also you learn to look slightly sideways at the stars to avoid  the situation illustrated below:
Extract and image source: BiologySE

When there is little light, the color-detecting cone cells are not sensitive enough, and all vision is done by rod cells. Cone cells are concentrated in the center of the eye, whereas rod cells are very rare in the center (image source):


From Rod Cells

Rod cells, or rods, are photoreceptor cells in the retina of the eye that can function in less intense light than the other type of visual photoreceptor, cone cells. Rods are concentrated at the outer edges of the retina and are used inperipheral vision. On average, there are approximately 90 million rod cells in the human retina. More sensitive than cone cells, rod cells are almost entirely responsible for night vision. 

And again as Jon says, you simply pay more attention as your eyes adapt. Experienced amateur astronomers learn to use red light and to avoid looking directly at a star, especially a dim one.
