1) If the company's offerings and "about us" are clearly in tune with what I as a customer am deliberately seeking, then I would stay.

The quicker a visitor leaves, the quicker they have ascertained that the company's services or products are not a "perfect fit" for what they are originally seeking.

How does a visitor decide?
A) through general look-and-feel
B) reading the "About Us" page
C) specific details on the product page is not what they are originally seeking (e.g. I am looking for a specific Reiko Kaneko Lip Tease mug in silver, but your site only has it in gold)
D) I would definitely stay on the website to make a transaction if someone I trusted directed me to a specific part of that website (because they knew me and knew what I was seeking --- referral trumps all)

2) I would leave in 10 seconds if the product, company, or offerings clearly do not fit my specific search criteria (price range, details, location, or just general gut feeling of "I don't like it")

As a web developer, I always tell my clients to specify, specify, specify. You can become extremely specific about what you offer, how you offer it, why your product/service is unique and special, etc. You can speak directly to your specific client -- you do not need to be as "generic" or to write very generalized descriptions.

If you try to speak to everyone, you end up speaking to no one.

State your message, vision, values, brand, manifesto in your terms and clearly differentiate how/why your company/organization is unique, as well as who you serve.

Bonuses on keeping people on the website:
if i) the price is clearly marked and
ii) there are easy-to-read headings, strong tags, & bullet points,
iii) no mandatory signup to review pages/products
iv) there's a mission or slogan or tagline at the bottom of every page.
v) contact information clearly marked, with whatever certifications are relevant to your industry
vi) blog is up-to-date
vii) "newish" feel and site is maintained: no spelling errors, grammar issues, missing images, 404 missing links