Wow, this is really a very complex question, multi-leveled.
IMO, retired = not having to show up to do any specific activity each and every week day (or some part thereof), for the purpose of being paid money.
So any voluntary w*rk, hobbies, ...etc, that you are doing NOT for the primarily purpose of making money (even though you may receive money) does not disqualify you from retiree status.
W*rk for the expressed purpose of making money (supplement your 'retirement' from your original j*b) means you have picked up another career and you are not retired (ex. cop who 'retires' and gets a j*b as a security guard to supplement income).
The middle ground is where you have an active hand in an investment like rental properties. I would think that if your primary j*b before your retirement was other than handling your rental properties and you have not increased your activity after retirement (like doing more of the maintenance and more active in managing the property), then you are retired. If on the other hand, you 'retired' only to make property management your new full timeactivity, then you switched careers.
In any case, after having stated that, it really doesn't matter. You are retired if you SAY you're retired. Who cares what other people think?
Now that is REAL freedom. ... free at last, free at last ...
btw, I totally agree with 'day of the week' and 'wearing a watch' as clear indicators of whether or not you are retired. I have to buy a newspaper or sign on to the internet to see what day it is. My self-winding watch runs down regularly and my DW laughs at me when I put it on (going out and she says I should wear it) and start waving my wrist back and forth for 10 minutes.