Does Kasisky test for Vigenère cyphers assumption always hold?
There is imprecision in what is stated in your notes. The Kasisky test only works if the corresponding letters in the two segments are separated by a distance that is a multiple of the key length (in other words they are encrypted by the same letter in the key). For example (source Wikipedia):

[abcdea]bcdeabcdeabcde[abcdea]bcdeabc
[crypto] is short for [crypto]graphy.


Note that the key here is $5$ letters long and the distance between corresponding letters in the two segments is $20 = (5*4)$ so the corresponding ciphertext segments will also be the same. It is also worth observing that there is no need for $l \le L$ as seen in the above example.