Is there a way for a TLS client to inform the server that it doesn't intend to resume its authenticated session(s) anymore?
Yes, by sending an Alert with a Fatal severity.
From RFC 5077:
5.1. Invalidating Sessions
The TLS specification requires that TLS sessions be invalidated
when errors occur.
This is an apparent reference* to the following text from RFC 5246 (emphasis mine):
7.2. Alert Protocol
One of the content types supported by the TLS record layer is the
alert type. Alert messages convey the severity of the message
(warning or fatal) and a description of the alert. Alert messages
with a level of fatal result in the immediate termination of the
connection. In this case, other connections corresponding to the
session may continue, but the session identifier MUST be invalidated,
preventing the failed session from being used to establish new
Practically speaking, choosing to terminate the SSL session with a fatal alert instead of with a warning close_notify alert probably won't cause anything worse than a log entry on the server. Because you're not closing until after your data exchange has concluded, it's unlikely the fatal alert will affect that. It's the moral equivalent of terminating a TCP connection with a RST instead of a FIN.
(I think you can actually just send a close_notify alert with Fatal severity instead of Warning severity, but I'd want to test that on the wire first).
Of course, if you have enough control over your client to impact how it closes the session, you have enough control to not send prior Session IDs on subsequent connections, which will achieve the same goal. The server will never try to resume a session unless the client has first proposed that Session ID. But if instead you're worried about third parties somehow gaining and re-using your Session ID, then triggering the server to purge it might be reasonable.
* When I say 5077 (January 2008) references 5246 (August 2008), it really was referencing an earlier version of the TLS spec; I'm just dragging out the most recent spec for you to look at. In fact the language for "7.2 Alert Protocol" is equivalent for TLS 1.1 (RFC 4346) and TLS 1.0 (RFC 2246).