First, for your code, besides changing predicted to new_predicted. You forgot to change the label for actual from $0$ to $-1$.

Also, when we use the sklean hinge_loss function, the prediction value can actually be a float, hence the function is not aware that you intend to map $0$ to $-1$. To achieve the same result, you should pass new_predicted to the sklearn function.

import numpy as np
from sklearn.metrics import hinge_loss

def hinge_fun(actual, predicted):
    # replacing 0 = -1
    new_predicted = predicted - (predicted == 0)
    new_actual = actual - (actual == 0)

    # calculating hinge loss
    hinge_loss = np.mean([max(0, 1-x*y) for x, y in zip(new_actual, new_predicted)])
    return hinge_loss


# case 1
actual = np.array([1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 1, 1, 0, 1, 0, 1])
predicted = np.array([0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1])
new_predicted = predicted - (predicted == 0)
print(hinge_loss(actual, new_predicted)) # sklearn function output 0.4
print(hinge_fun(actual, predicted)) # my function output 0.4

# case 2
actual = np.array([1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1])
predicted = np.array([0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 1])
new_predicted = predicted - (predicted == 0)
print(hinge_loss(actual, new_predicted)) # sklearn function output 0.5
print(hinge_fun(actual, predicted)) # my function output 0.5

Hinge loss is difficult to work with when the derivative is needed because the derivative will be a piece-wise function. max has one non-differentiable point in its solution, and thus the derivative has the same. This was a very prominent issue with non-separable cases of SVM (and a good reason to use ridge regression).

Here's a slide (Original source from Zhuowen Tu, apologies for the title typo):

Where hinge loss is defined as max(0, 1-v) and v is the decision boundary of the SVM classifier. More can be found on the Hinge Loss Wikipedia.

As for your equation: you can easily pick out the v of the equation, however without more context of those functions it's hard to say how to derive. Unfortunately I don't have access to the paper and cannot guide you any further...

1. The shape of the numpy array must be the same.

2. Each library has a different way to calculate the loss function. Therefore, the CS231 hinge loss and Scikit-Learn hinge loss formulas are different.(e.g. Keras sigmoid function)

3. The function with the same value as the Scikit-Learn function is as follows.

def hinge_loss(y_true, y_pred,  sample_weight):
    margin = y_true * y_pred
    loss = np.maximum(0, 1 - margin)
    return np.average(loss, weights=sample_weight)
    
hinge_loss(np.squeeze(labels_test), np.squeeze(y_pred), sample_weight)