Saying that waterfall has one set of SDLC phases and agile methods have their own phases is incorrect.
Both plan-driven and adaptive methods are ways of looking at systems development. Regardless of which methodology you are using, you are going to do the same thing - you're going to initiate an effort or team, you're doing to define some concept of what it is you are developing, you're going to plan out work, you're going to elicit and manage requirements, you're going to design solutions, you're going to integrate and test your solutions, you're going to deploy solutions and maintain them.
The difference between plan-driven methodologies and the agile or adaptive methodologies is when these things occur. The agile methods acknowledge the fact that it's not possible to fully plan out and specify a project up-front. By using short iterations and incrementally building a solution rather than creating a plan up-front and executing a project against it, you are able to adapt to changing needs.
Even characterizing Agile as "miniwaterfalls" is likely to be incorrect. The idea of a waterfall is that you, early in your project, have a fixed set of requirements. Taking Scrum as a specific example, your Sprint backlog isn't fixed at the beginning of a Sprint. You discover work, you negotiate with the Product Owner as new information is learned, and you work toward goals. You don't have a fixed set of requirements and design up-front and you don't have a test and integration period at the end.
The Agile methods are a different perspective on managing the work that goes into a project and a different set of principles that have been found to, generally speaking, work well when delivering software (and other types of products or services, in some cases).