UPDATE:
This is now live:

In general, we like the idea of changing the text to be a bit more informative. There's a lot of good ideas between the question and Tinkeringbell's answer. I poked at it a bit to try and find something that met the key points in the question while also being informative and firm.
One thing I ran into often on Interpersonal Skills is that moving the comments to chat doesn't always (or even often) cause the discussion to end. As a mod there, I often added big bold headings to make them stand out and be clear - don't keep posting comments here!
While I didn't want to do that since it's not usually needed, I did want to make it a bit more clear that continuing the existing comment discussion should not be an option without making any clear threats - at least y'all have comment locks, which I didn't as a mod.
To that end, I came up with this:
The comments have been [moved to a chat room] because they were being used outside their [intended purpose]. Often when there are lots of comments, they should be taken to [chat], become a [meta] discussion, or may be better served as an answer. Please consider whether one of these is a better option for this discussion but do not continue it here.
- "Moved to a chat room" links to the room
- "intended purpose" links to the comment privilege page
- "[chat]" is a magic link that will turn into "[sitename] Chat"
- "[meta]" is a magic link that will turn into "[sitename] Meta"
In this variant I'm leading with the action (comments were removed) rather than citing broken rules. This often can lessen the harshness of a statement... and then I explain generally why along with linking to the chat privilege page. I then answer the inevitable "so what do I do instead?" question and give three options. Finally, I politely ask them to go elsewhere to continue the discussion.
I tried to avoid any specific statements about whether or not to use the chat room - it feels odd to move comments to chat and then tell someone "but don't use this room I just created". This variation leaves that somewhat open ended.
This is not finalized - yet. I'm going to give it until the end of the week for feedback before I send it off to get the text changed - if you have any recommendations, I'm open to them.
Update
This one got put on a shelf for a bit, which I apologise for. After reviewing the recommendations in the comments and discussing a bit with JNat, we've decided to go with the following text, which is based off a recommendation from terdon:
Comments have been [moved to chat]; please do not continue the discussion here. Before posting a comment below this one, please review the purposes of comments. Comments that do not request clarification or suggest improvements usually belong as an answer, on [Sitename Meta], or in [Sitename Chat]. Comments continuing discussion may be removed.
The markdown will look like this:
**Comments have been [moved to chat](URL of chat room); please do not continue the discussion here.** Before posting a comment below this one, please review the [purposes of comments](/help/privileges/comment). Comments that do not request clarification or suggest improvements usually belong as an [answer](/help/how-to-answer), on [meta], or in [chat]. Comments continuing discussion may be removed.
I'll post another update once the changes are live.