I have came up with a term for the company I want to make. Please read it's company not a Product URL. The names that usually came in my mind were:
Do you think will it work?
About my company or products I want to make: I have a goal/vision that I want to make stuff that make people to become what I want them to become. For instance if the company name is Smart then I would like to make it getSmart because my products will make people to get smart.
Update: The URL will be getSmart.com but company's name will only be Smart so it will be used on Logo/Site/Letterheads etc
What do you suggest?
Based on the updated version of the question, I would suggest better not to make things complicated, at least not intentionally.
Keeping two versions of company identity has cost, and it's not cheap. Imagine you explain it to everybody you met, "well, actually our company name is Smart, but we want you to Get Smart, so Smart and GetSmart are all us". I bet you will get tired soon :)
For the examples you listed, I think the main consideration is the availability of domain. Squareup.com may want to acquire "square.com", but the single word domain may cost millions easily if not tens of millions, and is not always available for sale. They have to take a related alternative.
For your case, if you can acquire smart.com at affordable cost you definitely need to do that. If you can't, you can still go with GetSmart.com but identity yourself as Smart. No branding for GetSmart. Domain only.
Another important consideration is, before you brand yourself as "Smart", better to make sure the trademark "Smart" is not taken in your industry. If available you may need to consider applying for it once decision made.
With the word of "Get" you are indicating an intent. The intent for the user to take possession of something that is downloaded from your site. Hence, "GETfirefox.com". If you were to use just Smart. I'm sure that you would have branding and trademark issues in a dozen countries 50 different ways. Do some additional research.
Whether you add something at the beginning or the end of the core word, you are going to create some brand confusion if your domain name is different from your actual company name. Companies obviously can work thru these kinds of problems - Basecamp, Square, Fuze (meeting) come to mind. And there are also plenty of companies that append 'co' or 'inc', etc. to their domain names. But, imo, it is something to be avoided, if possible. Personally, I would work pretty hard to find a way to make the domain name and the company name match by morphing the company name to something unique which is both available and retains the original concept of the core word.
Assuming 'smart' is the core word you need to build around, I would tend to add something at the end of the name vs the beginning. This ensures that your brand is first in line in the domain name which means it is first in line in the minds of your customers/prospects.
There are several ways to do this: