With Esix's second ability, you can choose any creature you control besides Esix, even if it can't be the target of certain spells or abilities (e.g., because it has protection from creatures). (In the case of Teysa's triggered ability, however, you first destroy the creature, then create a token, so that with Esix, you can't choose the creature destroyed this way [C.R. 608.2c]. See also your other question involving Oversimplify.)
Esix's second ability isn't a triggered ability; it's a static ability with a replacement effect that can change what happens when you create tokens. Note that it lacks the word "when", "whenever", or "at". (See also Clone. It would be different if Esix had said, "Whenever you create one or more tokens for the first time each turn, you may choose target creature other than..." — compare Esix with Akim, the Soaring Wind.)
Esix's second ability isn't a triggered ability; it's a static ability with a replacement effect that can change what happens when you create tokens. Note that it lacks the word "when", "whenever", or "at". (See also Clone. It would be different if Esix had said, "Whenever you create one or more tokens for the first time each turn, you may choose target creature other than..." — compare Esix with Akim, the Soaring Wind.)