LIS with the wrong state definition

This function is supposed to return the length of the longest strictly increasing subsequence. It returns 4 for [10, 9, 2, 5, 3, 7, 101, 18] (correct) but 5 for [3, 1, 2, 5, 4, 6] where the answer is 4 (1, 2, 4, 6 or 1, 2, 5, 6). Find the bug.

1def length_of_lis(nums):
2    n = len(nums)
3    if n == 0:
4        return 0
5    # dp[i] = length of the longest increasing subsequence within nums[:i+1]
6    dp = [1] * n
7    for i in range(1, n):
8        dp[i] = dp[i - 1]
9        for j in range(i):
10            if nums[j] < nums[i]:
11                dp[i] = max(dp[i], dp[j] + 1)
12    return dp[n - 1]

1. Read the comment above dp. Is the recurrence consistent with that definition? Is the answer extraction consistent with it?
2. Trace [3, 1, 2, 5, 4, 6] and show where the value first becomes wrong.
3. Decide which state definition you want — "best ending at i" or "best within prefix i" — and make the recurrence, the initialization and the answer extraction all agree.
4. Write the corrected function and give its complexity. Mention the O(n log n) alternative.