The explanation on atomic size
Both of those explanations include part of the puzzle, but are not really complete on their own. To get a better understanding, you need to combine them, as well as understand something called Coulomb's law.

The basics of Coulomb's law are this:


The closer two oppositely charged things are, the stronger the attraction
The stronger the charge on either of the things, the stronger the attraction


With those points in mind, let's combine it all. I'll keep my explanation similar to the way you phrased yours:


Each step across a row on the periodic table adds both one nuclear proton and one valence electron. So, that is increasing both the total positive charge of the nucleus and the total negative charge of the valence shell, but not the distance between them (because electrons are added to the same shell).
The force of attraction between two charged objects depends on how strong each of them are (Coulomb's law). Since both the nucleus and the valence shell are stronger than in the previous step, and the distance between them has not increased (the electrons are added to the same shell) there is a stronger force of attraction between them, and the two are pulled closer together.
So, the atomic size of elements decreases across a period.