Backseating educational popup/warning
Adding new features that punish backseating is shortsighted and ignores that many users simply don't know better yet and would behave differently if they knew. A non-punitive, private reminder popup about backseating should be issuable to individual chat users or to the entire chat as a reminder. To describe the rationale more fully, here's a breakdown of the problem and a proposed solution.
Problems:
1- Many users don't know what backseating is, or do not completely know every behavior the term covers. Many users simply engage the chat and only discover later that what they said constitutes a backseating problem. This user experience is already poor. Making matters worse, the term "backseating" is rarely explained the same way twice. Users attempting to learn in good faith what it means might learn different standards for what behaviors are unwanted ("don't give the streamer gameplay commands and don't spoil the game"), only to discover there was more to the definition at a later date. For instance, it often also includes foreshadowing ("i love this next boss," "good luck with this next part"), discussions of the lore and so-called "fun facts" ("fun fact, the creator wrote this part of the story after robbing a real bank"). At present, Twitch seems not to offer any agreed-upon or official definition of this behavior on their platform to educate users that this behavior is sometimes unwanted.
2- Streamers' rules (chat popup, rules section, overlay prompts, etc.) about backseating are not always understood and often are ignored. For starters, if users don't know what backseating is, this term means nothing in a list of rules or reminders, meaning the behavior is just as likely to occur as ever. Users again are likely to engage the chat however they wish, only to learn later that it was a problem. Additionally, users might not even read rules where they currently appear anyway. Some streamers request in their rules that users type specific string of text in chat if they have read the rules. Anecdotally less than 1% compliance is common for new chatters, particularly if chatters arrive in a raid in a hurry to enter a raid message. Backseating behavior might be reduced for users that attend these rules, but more can be done to encourage acknowledgement.
3- Streamers and their moderator teams routinely have negatively experiences due to pervasive backseating while playing popular or classic games. It visibly upsets many streamers because their boundaries are being ignored, stories they hoped to have genuine first-time reactions to are spoiled, and left unchecked, chat messages between users begin to build on each other as a contest to establish who knows the most. Sometimes the expectation backseaters alone is enough to dissuade streamers from streaming certain games. Twitch can do much, much more to protect its creators who are earning them engagement.
4- Users who backseat in streams do not encounter helpful in-the-moment examples of alternative ways to interact with the streamer other than backseating. Said another way, if backseating is unwanted, specifically how should they act instead? Often backseaters did not intend to hurt anyone and approached a streamer playing a game they have fond memories of playing. Once reprimanded with current methods (callouts in chat by other users, callouts from the streamer, timeouts, or bans) backseaters typically feel unwelcome and leave without a good learning opportunity to engage the chat in a different way. They remain likely to repeat this pattern in other chats after they leave.
Suggestion to address these problems, which addresses problems 1,2,3,4
A- Create the ability to issue a reminder popup either for specific users or to all users in the chat, which requires them to acknowledge the content before they can participate in chat again. The reminder popup contains a Twitch-approved definition of backseating, examples, and alternative ways to chat. It should also contain optional links to video examples that load in a separate tab. This is likely to be more effective than requiring the streamer or mod team to use heavy-handed methods like timing a user out, banning them from chat, or verbally reprimanding them in front of the whole chat, which can negatively affect the stream engagement and community.
B- If users are shown this warning in multiple different chats over a period longer than a week, a Twitch reminder should be sent by email to warn them of the possible consequences if they continue to ignore this warning. Also include links to sources that model more acceptable chat behavior. The goal isn't to lock accounts or encourage a ban-evasion, it is to educate about this important boundary that many streamers require to participate in their communities.

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OwlysaurusRex commented
Much of this suggestion is being addressed with an upcoming moderator messaging feature. What still remains is that Twitch should include educational resources to help mods maximize the effectiveness of their messaging.
As mentioned here, merely saying "Don't backseat" means nothing if a person doesn't know what that means, and so the unwanted behavior is likely to continue.
Further, citing a chat rule doesn't tell the offending user how to behave instead. Education is needed on how to reframe messages so that they aren't offensive. Being able to link to articles or videos or even simply quick copy/paste examples would go a long way.