As an example of a set, I make the following notation: X = {1,2,3}
For comparison, I take the formulas of reflexivity and symmetry.
The first formula should be understood as (as I think):
For each x that belongs to the set X, the relation must contain a pairs (x, x) ∈ R. That is, the meaning is this: no matter how many elements we have in the set X, the reflexive relation is only that which contains a pair for each element X in the relation R, therefore R = {(1,1)} - not correct, but correct R = {(1,1), (2,2), (3,3)}
The second formula should be understood as (as I think):
For each x and y that belong to the set X, the relation should contain pairs that are inverse to each other (x, y) ∈ R => (y, x) ∈ R. By this conclusion, a relation is drawn in my head approximately as with a property of reflexivity, but with the rules of symmetry, that is, the wrong relation will be: R = {(1,1)}, but correct is R = {(1,1), (1,2), (1,3), ( 2,1), (2,2), (2,3), (3,1), (3,2), (3,3)} - all Cartesian product.
What am I misunderstanding?
For reflexive, it must contain all the pairs $xRx$, so your last example is correct. It can contain any other pairs desired, so $\{(1,1),(2,2),(3,3),(2,3)\}$ is reflexive as well.
For symmetric, note that it says $aRb \implies bRa$. If $aRb$ is not true it does not impose a requirement. Both your examples are symmetric, as is the empty relation. Another that is symmetric is $\{(1,2),(2,1),(3,3)\}$