First we need to define what is a tragedy, and differentiate its meaning from that of a domestic tragedy, in order to give Thomas Heywood's A Woman Killed with Kindness its due analysis.
In drama, a tragedy is a play or a composition that will present a situation that is going to cause disgrace, ruin, or pain onto the main characters. The thematic line of tragedy usually includes the inevitability of fate, karma, or any other issue that renders the main characters unable to fight what is coming to them.
The domestic tragedy is basically a less sophisticated version of what we know as the classic dramatic tragedy, because it brings the main idea of the play down to our level of reality. In domestic tragedies you do not get kings, queens, nor sumptuous and complex dynamics like you would in a classic drama. Instead, everyday people are often the main characters in these productions. Also, you have scenes depicting the typical dynamics of life, such as having dinner, cleaning a house, and other less than glamorous and everyday situations.
A Woman Killed with Kindness is set in the 17th century Yorkshire countryside, and its main characters are a superb model of a lady, Anne, and her husband, Sir John Frankford, who is a country gentleman. Anne is the catalyst of tragedy in the play because she falls for another man, and is caught. As a result, her virtue, the honor of her family, her lifestyle, her husband's ego, and life (as she knows it) are affected negatively since she is banished from the family, and her kids, for good. That would definitely classify this play as a tragedy.
The aspect of domestic tragedy that is evident in this play is that Heywood concentrates in setting up very common scenes of everyday life in the country. This is one of the most salient elements of this play to be considered a domestic tragedy. Moreover, the fact that the characters are every day people suffering from a real, yet sad, situation that could happen to anybody also make it a good example of domestic drama.
Finally, since the ending is tragic and people's lives are sacrificed either in life or death, it is safe to conclude that A Woman Killed with Kindness is, indeed, an example of a domestic drama, by every parameter.