Is there a formal definition of `nature` in Biology?
Is there a definition of "nature"?

I don't think there is any commonly accepted definition of "nature" in biology. To my experience, the term "nature" is actually relatively rarely used in conferences or peer-reviewed papers. 

Why is there no formal definition of nature?

The concept of "nature" has been developed outside the field of science or philosophy. As often concepts in the popular culture are used without a proper, complete definition. Even if there is an vague intuition that correspond to a concept, it does not mean that there is any way to make an objective definition of this concept. I think the absence of formal definition of nature mainly come from the fact that nature as used in the population culture does not mean much.

Is the term "nature" used in Biology and with which definition?

I just did a quick review of the use of the term "nature" in peer-reviewed articles. It seems that the term "nature" is rarely used or only in the abstract r in the first two paragraphs of the introduction to convey very general (and somewhat inaccurate) ideas.

Often, the term "nature" holds for "essence", "origin" or "intrinsic characteristic" rather than referring to "outside the lab", "landscape" or "ecosystems". Here are some examples of where I found the term nature 

"Nature" as "Essence"

From the abstract of Gibson and Dworkin 2004


  [..] we highlight recent progress in determining the nature and identity of genes that underlie cryptic genetic effects [..]


From the first paragraph of Woolhouse et al. 2002


  Failure to recognize the dynamic nature of the interaction could result in misinterpretation [..]


"Nature" as "outside the lab"

First sentence of the second paragraph of Elena and Lenski 2003


  Since Darwin’s day, many examples of evolution in action have been studied in nature [..]


Note btw that there is no good (objective) definition of life either. One might want to have a look at Why isn't a virus alive? for a discussion on the definition of life.