Andiamo works with many small businesses, organizations and start-ups, and we typically start out by developing their branding. The question inevitably comes up as we begin logo work - does my logo need to include LLC? does my logo need to include Inc? Ltd? The answer is a resounding "No." Nothing makes you look more rinky-dink than making a mistake like that.
Take a look at some logos from big national brands - none of them include Inc - this is because a logo is simply an image that represents the company, it is not the legal name. Logos depict the "trade name" of the company, and there is no legal reason to include Inc, LLC, Ltd, or any other legal designation on your logo.
Much like an individual's legal name - for example Mary Jane Smith III - the full name is used on legal documents, but not necessary anywhere else. Mary would use her full name on a mortgage, driver's license, marriage license, etc., but she needn't write out her full name every time she signs her name.
Much in the same vein, your company or organization's legal name identifies the kind of company you have registered for legal purposes, and should be used on all legal documentation for your company. Use the full legal name on any important documents - invoices, contracts, tax returns, legal records, etc. Your logo on the other hand is a separate entity for which you can use your shortened trade name, and display it throughout your marketing materials as part of your branding efforts, sans any legal designation.