What is the limiting reactant, and why I am I getting a contradiction?

Consider the burning of hydrogen gas using oxygen to produce water:

Balanced: 2H2 + O2 ----> 2H2O

If it was somehow possible to put an equal number of H2 and O2 molecules in the system. So 2 of each,

I would get 2 molecules of H2O produced with 1 molecule of unreacted oxygen. Since oxygen is in excess then H2 must be the limiting reactant.

But if I start with 1 gram of each, from 1 gram H2 I get 9 grams H2O; whereas from 1 gram O2 I get 1.125 g H2O. Therefore, I get less product from O2 so oxygen should be the limiting reactant.

Calculations:

1 g H2 * (1 mole H2 / 2 g H2) * (2 moles H2O / 2 moles H2) * (18 grams H2O / 1 mole H2O) = 9 grams H2O

1 g O2 * (1 mole O2 / 32 g O2) * (2 moles H2O / 1 mole O2) * (18 g H2O / 1 mole H2O ) = 1.125 g H2O

2 molecules H2 * (2 molecules H2O / 2 molecule H2) = 2 molecules of H2O

2 molecule O2 * (2 molecules H2O / 1 molecule O2) = 4 molecules H2O.

Summary:

When I use an equal number of molecules of each reactant,

I get less molecules of H2O from H2 so I conclude H2 is the limiting reactant.

When I use an equal number of grams of each reactant, I get less mass (g) of H2O from O2 so I conclude O2 is the limiting reactant.

Did I somehow choose a mass in grams of O2 which is an exception to the rule?