Difference in stellar abundance numbers

According to Lodders (2003, https://arxiv.org/pdf/1010.2746 ) the relative abundance of helium to hydrogen is $A({rm He})=10.925$, on a logarithmic scale where the hydrogen number abundance is 12. So this would mean a helium to hydrogen ratio, by number, of $10^{10.925-12}=0.08414$. i.e. 8.4% (your source uses 10.93, not 10.99, hence a very slightly different percentage).

Why have the two systems? The appearance of the spectrum - absorption lines, emission lines etc. depends on their number density. So it is relative numbers of the elements that count. On the other hand when you are doing structural calculations on stars, it is usually the mass fractions that matter.

The relationship between the two?

The relative numbers of helium and hydrogen are easily translated into relative mass fractions of helium ($Y$) and hydrogen ($X$): $$frac{Y}{X} = 0.08414 times 4.0026/[(1-0.08414)times 1.0078] =0.3649$$, where $4.0026/1.0078$ is the ratio of atomic masses of helium to hydrogen.

Now there is also a small fraction by mass of heavier elements $Zsimeq 0.014$, where $1 = X + Y +Z$.

If we substitute $X= Y/0.3649$ and $Z=0.014$, then $$ Y left(1 + frac{1}{0.3649}right) = 1-Z$$ $$ Y = 0.264$$

To sum up: $$Y = frac{1-Z}{left(1 + frac{1.0078(1 - 10^{A({rm He})-12})}{4.0026times 10^{A({rm He})-12}}right)} = frac{1-Z}{left(0.7482 + 0.2518times10^{12-A({rm He})}right)}$$