A convention is an officially-designated assembly for a special purpose.
A convention of the states is a kind of convention.
A convention for proposing amendments is a kind of convention of the states.
The broadest term is convention. In constitutional use, a convention is an official meeting or task force called to address assigned issue(s). Don’t confuse this constitutional use with unofficial meetings of civic groups or political parties, such as the “convention” of the American Medical Association.
The Constitution assigns specific tasks (which the courts call “federal functions“) to conventions, just as it assigns specific tasks to state legislatures, state governors, grand juries, and trial juries. For example, the Constitution assigns amendment-proposing powers to an interstate convention and amendment-ratifying powers to in-state conventions.
The Constitution assigns election of the president and vice president to a compound (in-state/interstate) convention we call the Electoral College.
Convention of the States/Convention of States. These terms are synonyms. Both arose before the Constitution was adopted.
Conventions are either in-state or interstate (federal). An in-state convention sometimes is called a convention of the people. An interstate or federal convention is a convention of the states.
In a convention of the states, each state meets on terms of sovereign equality. Each is represented by a delegation of commissioners chosen as the state legislature directs.
Like basic jury protocols, basic convention of states protocols were fixed long before the Constitution was written. Conventions of states borrowed them from international law. From 1677 to 1776, there were 20 conventions of colonies. From 1776 to the present, there have been at least 20 conventions of states.
A Convention for Proposing Amendments is a kind of convention of states. The Constitution’s framers selected this device for a special purpose, just as they selected juries and state legislatures for other purposes. The Constitution thereby adopted the traditional protocols and definitions for all these assemblies.
For more detail, see The Law of Article V.