"when does one’s body start storing fat either by increasing the number of fat cells or by increasing the size of existing fat cells? "

The answer is;  increasing the size of existing fat cells

When there are excess amino acids, glucose, fatty acids and glycerol in the blood, insulin produced by the pancreas causes the cell to absorb and store them. Fat cell, liver, and muscle cells are especially specialized for this role. Fat cells also convert excess proteins and carbohydrates to fats (a process which is less efficient that absorbing and storing fats from the bloodstream). The more fat droplets they store the larger the cells become.