I've no problem with being described as 'middle-aged' or 'old', but I think the meanings of both have changed. Look at how people in their early to mid twenties are often now seen as veritable 'babies', in part I suppose because returning home to your parents after university has become normalised, and because home ownership/independent living/first job, which used to be 'signs' of adulthood, have all become problematised for many.
I only stopped being a student at 29, so only had my first real professional job at almost 30 (though obviously working long, long before that), and had my first child just shy of 40. That probably makes me 'younger' in some ways than someone who started working fulltime at 18 and had their first child at 20, and who could potentially have a grandchild the age of my toddler.
And I think changes in the retirement age make a big difference to what we can legitimately call 'old'. The boundaries are pretty fluid, but I doubt I'd consider middle age starting below fifty or so, and 'old age' is pretty nebulous as a concept.