5 USC 551(4) says that "rule"
means the whole or a part of an agency statement of general or particular applicability and future effect designed to implement, interpret, or prescribe law or policy or describing the organization, procedure, or practice requirements of an agency and includes the approval or prescription for the future of rates, wages, corporate or financial structures or reorganizations thereof, prices, facilities, appliances, services or allowances therefor or of valuations, costs, or accounting, or practices bearing on any of the foregoing
One can find "the final rule" published in The Federal Register on a frequent basis. The definitions section does not define "regulation". Now turning the page, 5 USC 552(4)(A)(i)
In order to carry out the provisions of this section, each agency shall promulgate regulations, pursuant to notice and receipt of public comment, specifying the schedule of fees applicable to the processing of requests under this section and establishing procedures and guidelines for determining when such fees should be waived or reduced
and that (6)(B)(iv) says
Each agency may promulgate regulations, pursuant to notice and receipt of public comment, providing for the aggregation of certain requests by the same requestor, or by a group of requestors acting in concert, if the agency reasonably believes that such requests actually constitute a single request, which would otherwise satisfy the unusual circumstances specified in this subparagraph, and the requests involve clearly related matters. Multiple requests involving unrelated matters shall not be aggregated.
There is also language about multitracking and requests for records, the point being that there shall be regulations about those operations of the government. From what I can tell, "regulation" in this section refers to rules about rule-making: they are rules about rule-making.
5 USC 552a(h)(i)(1) refers to prohibitions "by rules or regulations", which indicates that "rules" and "regulations" are different things. Subsection (u)(3)(A) speaks of the need to comply with "all relevant statutes, regulations, and guidelines", with no mention of rules, and from the context it appears that what's called for is compliance with rules on what agencies can do with data.
While the restrictions on what doctors can do, pursuant to HIPAA, or what schools can do, pursuant to FERPA, are commonly called "regulations", it appears to me that federal law actually distinguishes "rule" and "regulation", and while you and I may be subject to rules, only rule-making agencies are subject to "regulations". Is there any case law, rule, regulation, statute or legislation that clarifies the meaning of "rule" and "regulation" at the federal level?