I wouldn't say so, though you could easily argue the opposite. Loosely, to me, biotechnology involves the interaction of biological processes with technology - things like tissue engineering (though, this is also biomedical engineering!). To give an example of the opposite, I feel that things like the design and development of prosthetics & orthotics would count as biomedical engineering but not biotechnology.
I'm not sure I'd say genetic engineering is either, I'd probably just call it genetics as a sub of molecular biology. This all said, I don't really think the labels are all that important, though one tends to develop preferences for ones own description. I know people in medical physics/clinical science (whom are labelled clinical scientists or
medical physicists) who, nonetheless, market themselves as clinical engineers.