3. The Senate shall elect its leader and other officers, and shall have sole authority to determine its own methods of proceedings.
4.
The House of Representatives shall elect its Speaker and other officers, and shall have sole authority to determine its own methods of proceedings.
Pretty clear here, isn't it? If 5 representatives and 4 senators make a determination with regards to their own rules, they each have the sole authority over their own chambers and thus can change the rules in such a manner. I don't see anything here that would imply the threads need to be separated?
Your point about texts is interesting I suppose but I don't think it's particularly relevant; I'd argue that it's implicit here that the Senate is voting on repealing it from their own rules while the House is doing the same for their rules.