The Proper Use of the Stack Exchange Name section in the legal trademark guidance page says:
Stack Exchange Inc. is the official name of the company.
So, as per the given official name, it should be "Stack Exchange Inc." (notice the dot after Inc) and not "stack exchange inc".
Upon doing further research on the corporation (or the legal business entity) name, I found the following facts.
EDGAR (Electronic Data-Gathering, Analysis, and Retrieval):
EDGAR system performs automated collection, validation, indexing,
acceptance, and forwarding of submissions by companies and others who
are required by law to file forms with the U.S. Securities and
Exchange Commission (the "SEC").
EDGAR system search says: Stack Exchange, Inc.
NYS Department of State, Division of Corporations: (for some reasons info. is printed in "CAPITAL" letters)
The Corporation and Business Entity Database includes business and not-for-profit corporations, limited partnerships, limited liability companies and limited liability partnerships, as well as other miscellaneous businesses.
NY DOS database search says: STACK EXCHANGE, INC.
IMO, the correct way to write a company name is to write it as the company registered its own name. So, based on the above facts, the "stack exchange inc" should be changed to "Stack Exchange, Inc." (proper noun; notice the comma and dot). But, if the company's official doc. says "Stack Exchange Inc." then let's go with that at least and not with "stack exchange inc".
I completely agree with the comments posted (under the questions above) by Monica Cellio (elected ♦ moderator) and animuson (employee ♦ moderator).