I would suggest that Lean software development as discussed at length by Mary Poppendieck and Tom Poppendieck is a methodology closely related to the Agile family of software development methodologies such as Extreme Programming and Scrum, each with its own design emphasis, rituals and community. However, at the heart of all Lean/Agile culture is the core belief that rapid feedback cycles - where feedback is meant to be in the form of ‘live’ reactions (such as sales or usage spikes) from the ‘real environment’ (from clients, prospects and users most importantly) – are required to discover the important but false assumptions often hidden in planning or requirements documents. So you can think of the ecosystem of Lean/Agile practices as vehicles for enabling fast feedback cycles - ‘fail fast to succeed sooner’ – in order to expose and manage the risk associated with false assumptions about the development environment.
Lean Startup simply applies the idea of using rapid feedback cycles to expose false assumptions to the customer development and business side of the organization. Steve Blank coined the term and created the processes around ‘Customer Development’ which Eric Reis built upon to develop ‘Learn Startup’.
Kanban is a tool, like you mentioned, that can be used in any project management style, even Waterfall.