How can I properly calculate the isoelectric point (pI) of amino acids? The following amino acid is called lysine. I was asked to calculate its isoelectric point, with the given $\mathrm pK_\mathrm a$ values.

I've searched a lot, and the most helpful post that I found was How do I calculate the isoelectric point of amino acids with more than two pKa's?
According to orthocresol's answer:

Since the $\mathrm{pI}$ is the $\mathrm{pH}$ at which the amino acid has no overall net charge, you need to average the $\mathrm pK_\mathrm a$ values relevant to the protonation/deprotonation of the form with no net charge.

Let's call the ends $e_1, e_2$ and $e_3$ (from left to right).
Approach $\#1$

deprotonate $e_3$ (i.e., carboxyl group)
deprotonate $e_1$ or $e_2$ [neutral point]

So, $\mathrm pK_\mathrm a$'s of $e_1$ and $e_2$ are relevant.
$$\Rightarrow \mathrm{pI} = \frac{10.53 + 8.95}{2} = 9.74$$
But, is there some limit to number of protonations/deprotonations or some procedures to follow?
For instance,
Approach $\#2$

deprotonate $e_1$
deprotonate $e_3$ [neutral point]
deprotonate $e_2$ and protonate $e_1$ [neutral point]

This time, $\mathrm pK_\mathrm a$'s of $e_3$ and $e_1$ are relevant. But, the calculated $\mathrm{pI}$ isn't correct.
So, how can I validate the approaches?

Problem source: FIITJEE study material
$\mathrm pK_\mathrm a$ and $\mathrm{pI}$ values table for amino acids: https://www.anaspec.com/html/pK_n_pl_Values_of_AminoAcids.html
Isoelectric point of an amino acid is the $\mathrm{pH}$ at which the molecule carries no net charge[1]. It can be calculated by the average of the relevant $\mathrm pK_\mathrm a$ values as you have mentioned.
Your confusion seems to stem from choosing the relevant $\mathrm pK_\mathrm a$ values. For this we should refer to the titration curve of the amino acid.
For a neutral amino acid[2]:

From the curve we can infer that the $\mathrm{pI}$ is simply the average of the two $\mathrm pK_\mathrm a$ values of the carboxylic acid and the amino group.
For a basic amino acid[2]:

From the curve we can infer that the $\mathrm{pI}$ is simply the average of the two $\mathrm pK_\mathrm a$ values of the two amino groups. The $\mathrm pK_\mathrm a$ of the carboxylic acid group is not relevant.
For an acidic amino acid[3]:

From the curve we can infer that the $\mathrm{pI}$ is simply the average of the two $\mathrm pK_\mathrm a$ values of the two carboxylic acid groups. The $\mathrm pK_\mathrm a$ of the amino group is not relevant.
Here are examples for all three cases:

References:

Wikipedia

Titration curves for neutral and basic amino acids

Titration curve for acidic amino acids