I have always believed that the PM instructs the Contractor (under a Project Manager’s Instruction) however a colleague disagrees and believes that the PM notifies the Contractor (under a Project Manager’s Notification)
I think you have answered your own question here. You rightly say that the PM response can be to instruct a revised quotation. The word there was instruct, not notify.
I think your question here is more a case of “what piece of paper should they be using to confirm this” i.e. if you have separate proforma called a “PM Instruction” and a “PM notification” which one do they use? Franky it does not matter - the important thing is that they are confirming in writing and separately from any other form of communication that they want the Contractor to requote.