I think you need to look at this from one level higher. It’s not how the AD is written but what an AD is in the big picture. Every Airworthiness Directives is part of FAR Part 39 and requires a separate write up and sign off to show compliance, when applicable, regardless of its subject matter. That is the law. And since standalone SBs are not mandatory for 91 ops, but all ADs are, your sign off of only SB1309 does not substitute for that specific AD legal requirement. So no AD sign off by Feb 7 puts you in violation of the AD provided you "operate" the aircraft. Don't fly no problems until the AD is signed.
Also without a sign off of this AD in your aircraft record, you would be in direct violation of 91.417(v), plus your IA or mech could not sign off your next annual or 100hr as airworthy. There are a number of FAA guidance docs on ADs. Here is one from AC39-7D:
10. AD COMPLIANCE. ADs are regulations issued under part 39. Therefore, no person may operate a product to which an AD applies, except in accordance with the requirements of that AD. Owners and operators should understand that to “operate” not only means piloting the aircraft, but also causing or authorizing the product to be used for the purpose of air navigation, with or without the right of legal control as owner, lessee, or otherwise. Compliance with Emergency ADs can be a problem for operators of leased aircraft because they may not be aware of the AD and safety may be jeopardized.
You’re always entitled to your own interpretation, but as often quoted on this forum “if it’s not in the logbook it didn’t happen” this is the one example of not making the entry that will come back to bite.