The Example environment stems from the module "Theorems (AMS, Numbered by Type)" - defined in theorems-ams-bytype.inc - that you've included - :
Style Example
CopyStyle Definition
LatexName example
LabelString "Example \theexample."
Preamble
\theoremstyle{definition}
\newtheorem{example}{\protect\examplename}
EndPreamble
Requires amsthm
LangPreamble
\providecommand{\examplename}{_(Example)}
EndLangPreamble
BabelPreamble
\addto\captions$$lang{\renewcommand{\examplename}{_(Example)}}
EndBabelPreamble
LabelCounter example
End
Under Windows, this is located in
C:\Program Files (x86)\LyX20\Resources\layouts
The example counter used is defined in theorems-ams-bytype.module, located in the same folder. You'll need some understanding of LaTeX to modify this to your liking, including LyX' syntax.