Btw how channel closure procedure is initiated? I believe if either party want to close channel they don't really need other party to agree - they just need to post latest channel state to blockchain and it's done.
Maybe I'm telling you the obvious, but, a transaction is a "piece of data" that potentially can go into the block chain ; however, in order to get into the block chain, it must be sent to a mining node. That mining node will check its validity, and if it likes your transaction, it likes your fee, it might decide to include it in a block that it will try to calculate proof of work for, publish to other mining nodes, who may accept it and put their blocks on top of it etc...
In other words, a transaction, sent to a mining node, that can be included in the block chain, and hopefully will be included in the block chain, is a "broadcast" transaction.
In order for a transaction to be acceptable, it needs all the right signatures as specified in the output scripts corresponding to its inputs.
The whole idea of the LN network is that the users of a channel exchange transactions amongst them, but don't broadcast them. So they are not included in the block chain ; but anyone of them having these pieces of data could decide to broadcast them. However, given that these pieces of data are limited to the partners of the channel, only they have them, and only they can broadcast them. Nobody else knows them. On top of that, most of these exchanged pieces of data have still a missing signature. At every moment, each of the partners has the piece of data, ready to be broadcast, except that their own signature is missing. So at any moment, they can decide to sign that piece of data, at which point it becomes a genuine transaction, and can be broadcast to a mining node, that can include it in one of its next blocks, like just any transaction in bitcoin. That act of signing and broadcasting this transaction is exactly the "closure procedure".
Because from the block chain PoV, there was only a transaction when the channel was opened, from both partners' addresses to a specific address, then nothing, and then the closing transaction from that specific address back to the addresses of the participants.
The whole thing of the LN network is that the participants can exchange amongst themselves POTENTIAL, half-signed transactions that UPDATE the state of the channel in such a way that at no moment, someone can trick his partners, and that the sheer possession of these potential transactions is enough to guarantee that they can obtain their coins by settling if they want to.
This is slightly akin to both "placing their coins in a common vault", and then keeping the books of who owns what, without the need of getting the coins out of the vault, just updating the books, because each partner has the cryptographic guarantee that he can, at any moment, demand that the vault is opened, and that each gets their share as it was specified during the last update of the books.
The only thing with the "timing" is that nothing stops someone from settling by broadcasting a previous transaction that was the right balance a week back, but is not the right balance any more. One cannot "erase" data from other computer's memories: at one moment, that settlement was possible, and without doing anything, that remains still valid. So there must be a time frame to "contest the settlement" if it isn't the last one, by the other partner. Once that time is expired, the time to complain is over and the settlement is final.