Is this a valid implementation of the Selection Sort Algorithm?

I have tried to implement the Selection Sort Algorithm in two different ways, and I was just wondering if that is considered to be a valid implementation of that algorithm. Would that be also considered an efficient implementation of the algorithm, or are there different and more efficient ways?

Here are my implementations:

Implementation Attempt #1

def selection_sort(L: List[int]) -> List[int]:
    for i in range(len(L) - 1):
        j = i
        smallest_num = L[i]
        while j < len(L) - 1:
            if smallest_num > L[j + 1]:
                smallest_num = L[j + 1]
            j += 1
        L[L.index(smallest_num)], L[i] = L[i], L[L.index(smallest_num)]
    return L

Implementation Attempt #2

def selection_sort(L: List[int]):
    for i in range(len(L) - 1):
        j = i
        while j < len(L) - 1:
            swap_min(L, i)
            j += 1
    return L

def swap_min(L: List[int], n: int) -> None:
    smallet_num = min(L[n:])
    sm_indx = L.index(smallet_num)
    L[n], L[sm_indx] = L[sm_indx], L[n]

Many thanks in advance!