Is there some way to add a third party authorizer to (or by) a lightning network transaction? I'm not asking for "multisig on the lightning network", but I'm more curious about how one would set up shared control over lightning network transactions in general. 

Imagine we don't have a multisig transaction... would it be possible to set up some sort of 3rd party oracle that would "link" a channel? 

Or maybe set up some sort of logic implemented by the opening or closing of payment channels on a boutique lightning network? 

In 2.2 of the lightning network paper, it says that opening payment channels doesn't actually control the movement of any funds. 
https://lightning.network/lightning-network-paper.pdf

Some folks came up with an idea for a way to pay an oracle for provable information (especially information that can best be obtained in from a lot of different decentralized nodes at "anti-affinitied" locations) using lightning &amp; bcoin: https://github.com/michaelfolkson/bcoin-hackathon

My brother wrote an implementation of NAND gates here, https://github.com/rjb25/gate-wiring , and since considering both of these I wondered whether one might hook up some sort of gating system whereby a third party could make a decision about two possibilities for distribution of the funds of another two parties. 

And now I'm wondering if it'd be possible to set up some sort of private lightning network that uses the opening and closing of payment channels (or resizing them) to do some kind of third party authorization logic for lightning payments?

Perhaps where one person (party A) receives some amount of bitcoin in one case, and another (party B) in a different case, as determined by a third party oracle or mod?
I think you ask about escrow services in lightning. Generally there is a good opportunity to work with preimages. For example if we would move away from sha256 and would use ecdsa public private key encryption or a homomorphic hash function. Several parties could collaborative create a payment hash Wo sich they could only together generate the preimage. Stuff like this would also be possible with scriptless scripts. Also multiparty channels and channel factories could yield such functionality. So yes generally it is possible with lightning to go down this road but currently nothing of that is implemented to the protocol.