Determinant of a Matrix using Gauss Elimination, inconsistent answers

Question

I have worked through finding the determinant of the following Matrix

$$ \begin{pmatrix} 6 & -1 & 0 & 4 \\ 3 & 3 & -2 & 0 \\ 0 & 1 & 8 & 6 \\ 2 & 3 & 0 & 4 \\ \end{pmatrix} $$

and I have noticed that when I do gaussian elimination the way that I am used to, I always get the wrong answer. Now I don't want to sound cocky, but I don't believe I am performing the elimination incorrectly, as when I use the same method in other applications; solving linear systems of equations; it works out. Am I interpreting the rules of gaussian elimination to find determinants incorrectly?

For example, the first step that I would take for this question, and the general format that I do gaussian elimination would be to replace row 2 with $$ R_2 = 2R_2 -R_1 $$ In other words my general way of doing gaussian elimination is by taking the row I am going to be replacing, multiplying it by a multiple and subtracting it by the pivot row.

I have also attempted the problem a different way (after seeing some other gaussian elimination examples) and used this style $$ R_2=-\frac 12R_1+R_2$$

This latter style of doing gaussian elimination gave the correct answer, but that is my question, why? Are both methods not linear combinations? And I am aware of the 3 rules that govern gaussian elimination with regards to finding determinants that state

1. If you swap two rows you must multiply the determinant by (-)
2. If you multiply a row by a scalar you must multiply the determinant by that scalar value
3. The linear combination of rows do not change the determinant

So I have made sure to not use the first two rules to rule out any mistakes regarding that, so again I am left scratching my head a bit.

Thank you and sorry for the long write up.

I have also included pictures showing my full workings with both "styles"

Gaussian Elimination Using "my style"

Gaussian Elimination Using "proper style?"