Ask for their version before correcting it
Invite them to say what they heard, saw, or were told and what it meant to them. Listen for the exact point where your understandings separated.
Separate intent, facts, and impact
State what you meant and any verifiable context, then acknowledge the impact that remains. Avoid telling them they are irrational simply because your intent was different.
Agree on what happens next
Correct a rumor with the people who received it, clarify future communication, apologize for your contribution, or accept that trust needs time. Do not pressure mutual friends to take sides.