Yes, it is true, the proper emphasis should be on "life."
There is a common conceptual confusion regarding life and death, due in large part to the way in which we conceptualize life and death as opposite abstract poles, without taking into account that the two concepts are intimately intertwined over the passage of time.
Our current way of life, that is, the daily administration of industrial civilization in the form we currently have, with its attendant phenomena such as radical egotism, oil addiction, short-termism, and many others, is in fact the daily practice of death. It is a way of life that contains in its daily processes the seeds of our eventual destruction.
To truly embrace life, in respecting the full meaning of the word, is to acknowledge, mourn, and part ways with an old way of life, in other words, to embrace death, so that the limited space in which we conduct our collective affairs can be cleared to make way for the actual pursuit of life.