I've heard about Cantor's diagonal proof that it's impossible to count all real numbers between 0 and 1. However, isn't that proof dependent on the way in which the numbers get counted?

Every time I've seen the proof, the numbers have always been listed directly in ascending numeric order, and then the diagonal line shows that ever nth digit of every nth number is can be changed to make a new number.

However, just like it's possible to count all rational numbers (or fractions) by laying them out in a 2D grid and counting diagonally, isn't it possible to construct the sequence of all real numbers in a way that also makes them countable?

For example, I can list all reals with 0 decimal places, then 1 decimal place, 2 decimal places, 3, 4, and so on... Like this:

0.0  1.0

0.1   0.2   0.3   0.4   0.5   0.6   0.7   0.8   0.9

0.01  0.02  0.03  0.04  0.05  0.06  0.07  0.08  0.09
0.11  0.12  0.13  0.14  0.15  0.16  0.17  0.18  0.19
0.21  0.22  0.23  0.24  0.25  0.26  0.27  0.28  0.29
0.31  0.32  0.33  0.34  0.35  0.36  0.37  0.38  0.39
0.41  0.42  0.43  0.44  0.45  0.46  0.47  0.48  0.49
0.51  0.52  0.53  0.54  0.55  0.56  0.57  0.58  0.59
0.61  0.62  0.63  0.64  0.65  0.66  0.67  0.68  0.69
0.71  0.72  0.73  0.74  0.75  0.76  0.77  0.78  0.79
0.81  0.82  0.83  0.84  0.85  0.86  0.87  0.88  0.89
0.91  0.92  0.93  0.94  0.95  0.96  0.97  0.98  0.99

0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009
0.011 0.012 0.013 0.014 0.015 0.016 0.017 0.018 0.019
0.021 0.022 0.023 0.024 0.025 0.026 0.027 0.028 0.029
0.031 0.032 0.033 0.034 0.035 0.036 0.037 0.038 0.039
0.041 0.042 0.043 0.044 0.045 0.046 0.047 0.048 0.049
0.051 0.052 0.053 0.054 0.055 0.056 0.057 0.058 0.059
0.061 0.062 0.063 0.064 0.065 0.066 0.067 0.068 0.069
0.071 0.072 0.073 0.074 0.075 0.076 0.077 0.078 0.079
0.081 0.082 0.083 0.084 0.085 0.086 0.087 0.088 0.089
0.091 0.092 0.093 0.094 0.095 0.096 0.097 0.098 0.099

0.101 0.102 0.103 0.104 0.105 0.106 0.107 0.108 0.109
...

If I continue in this same pattern infinitely, wouldn't I then have counted all real numbers between 0 and 1?