There are strict definitions of the terms "animal welfare" and "animal rights," but there are other legitimate meanings.
Although an individual on either side of the question might try to limit a discussion of "animal rights" to a single, straightforward definition, the fact is that the term "animal rights" is often used informally to encompass a mixture of two or more meanings.
When the ASPCA defines The Five Freedoms (Farm Animal Welfare Council et al. 1979), it refers to animal welfare. But the question remains, what are people's motivations for providing The Five Freedoms? For some, it's their faith in the bible: when God gave man dominion over the earth, He gave man the responsibility to rule wisely and fairly. For others, creation stories portray man in relationship with his/her fellow creatures, ranging from relatives with shared destinies (Carroll 2013) to adversary (Bible: Genesis 3), to lord. Personal philosophies acknowledge that every living thing has "natural rights" (see quotes from Jean-Jacques Rousseau, Jeremy Bentham, Edward Nicholson, Arthur Schopenhauer, John Stuart Mill, etc.; e.g. in the Wikipedia article on "Animal Rights").
In other words, don't let some bully (e.g. NAIA 2016) convince you that "animal rights" is impossible because it wouldn't work in court. It works in life, and that's what matters.
Bible, Book of Genesis. Chapter 3.
Farm Animal Welfare Council, Farm Animal Welfare Committee, ASPCA, and RSPCA. 1979. “Five Freedoms of Animal Welfare.” American Society for the Prevention of Cruelty to Animals (ASPCA).
NAIA. 2014. What is Animal Welfare and why is it important? National Animal Interest Alliance (NAIA).
Wikipedia. 2016. "Animal Rights."