Web application for Software as a Service - Shared database or Multiple Databases?
This is the old "how to handle multi-tenancy" question for which there is no one right answer. You have covered a good selection of the key points. I can't provide an answer beyond "it depends" but here are a couple of other points for you to consider:


mixing data of different clients ... if there is a bug, its easier for someone to end up accessing some else's data and Come up with a scheme associating each installation [correct] database - the risks here are very similar if the various databases are on or can be accessed by the same hosts, especially from the point of view of a defending against deliberate attacker, so that part of the decision comes down to which collection of potential security concerns you feel you can efficiently design around most effectively. If you feel your SQL skills are lacking that might push you in the multiple database direction due to concerns about large data sizes become inefficient as your client base grows.
For each installation, buy a DB from the hosting company for a SaaS solution you have an inappropriate hosting arrangement if you are paying per database or similar. It sounds like traditional shared web hosting were you should be looking at dedicated machines (virtual or physical) or a PaaS based platform.
Consider carefully how you intend the application to scale. The multiple databases option gives you the freedom to split clients between database servers as your needs grow, but the single database option is easier to manage in a PaaS environment if you design is appropriate, letting the platform provider worry about the physical reality of scaling as you pay for more resources.
Also on the scalability matter, make sure consider your administrative responsibilities: how will you backup the data, restore parts of it if there is an error/accident, or restore all of it in the event of an application/platform wide catastrophe, and what timescales do you want/need to work to in such events?


For more information see https://msdn.microsoft.com/en-us/library/aa479086.aspx, https://en.wikipedia.org/wiki/Multitenancy, and search for similar terms for further discussion. (that first link is from MS, but covers the matter from a general design perspective rather than being specific to MS SQL Server, it has also been around a while but the matters concerned haven't changed as much as you might think the would have in the last decade)