Scrum and Agile Software Development

The point you raise over who comes first, the customer or the client is the classical split between large and small businesses.
When your dealing with smaller clients and you need to get paid its common practice to put the client first, even when they havnt got a clue about websites. Big companies however will put the customer first because they understand that the customer is truly what drives the business. Telling that to a client is hard though, especially when its the smaller clients who want to have a part in everything.
I sometimes use the line, am I making the site for you or the end users, sometimes it works, depending on the client. Larger clients are easier to deal with because you are working with managers who are experienced in that department and who will defer to your expertise instead of the business owner who is generally clueless but convinced that they have the knowledge required to countermand decades of experience.

As for taking you time and getting it right, generally I agree with making sure that you deliver a proper working product however the agile terminology is generally used with larger companies who are paying a lot of money for contractors who they expect to know all the best ways to do things already and understand best practices and techniques instead of learning on the job. For those companies, every day in development costs a lot in contractor fees but also costs even more in terms of potential revenue lost from not having an improved solution.

Also when I develop a site or a solution I like to hit the ground running and develop hard and fast, develop a working product while the code is fresh in your mind, get it made before external factors slow down the development cycle.
Eg: feature creep. If its a good feature you can normally implement it after and if its a stupid feature you can say that would require an extensive redevelopment. Get a working site made thats nice and simple with a good solid HTML framework and afterwards you can add bells and whistles. Pus if you give the end client a solution they will always want to make some last minute changes so its better that they ask for some fancy features on top of a simple solution than asking you to redesign an already complex solution.