I believe that you are mixing up the concepts of electronegativity and of electron affinity.
The first statement you present is related to Electronegativity.
If two similar sized atoms have different electronegativity values, the atom with higher electronegativity holds the electrons more tightly, and thus is more stable.
Yes. This is generally true.
Electronegativity establishes how well an element can pull electron density towards itself.
The second statement is about electron affinity.
If two atoms have different sizes (down the periodic table), the larger atom has electrons more spread out and thus is more stable. Example: fluoride vs iodide anion
Generally true as well. Flouride anion is less stable than iodide anion due to the repulsion of electrons in the outermost octet of fluoride. But this rule is not absolute as chloride has the highest stability (in the gaseous state) though it is above bromine and iodine.
Considering some confusion in the comments I shall put this here.
The electronegativity is generally not compared based on inter-electron repulsion because the shared pair doesn't completely enter the orbits of the electronegative atom but is shared and so electron density is between the two atoms and not concentrated on one atom for repulsion to take much effect.