There are MOSFETs with and without the body diode
Yes, albeit very rarely; and also, there are those with the substrate ("back gate") taken out to a separate pin (or.. there were). In that case, of course, the diode behavior is related to the substrate pin, and the "diode-ness" of the triangle remains relevant, just not automatically strapped to the source like most.
One example of current technology that doesn't have a substrate connection is EPC's eGaN FETs. From what I've seen in their appnotes, you can indeed run them in reverse: except that, because Vgs(max) limits how far you go, and Vgs(th) is low, they still kind of act like they have body diodes, in recommended operation.
The correct symbol in this case, could go a few ways: the "channel" could be a solid or broken line (I think the intended difference on that one is enhancement vs. depletion mode), with no triangle to indicate the substrate junction (because it doesn't exist), but, some means is still needed to indicate the polarity of the channel (P or N). The triangle nomenclature could still be used, but merely with no connection made to it (which is true).
and there are both N and P channel MOSFETs. The standard symbols do not show the body diode whether it is there or not and the only way to determine the MOSFET polarity is the inside triangle.
Like commonly used in CMOS IC schematics? I think they only do that as shorthand. The substrate connection is implied (to the respective rail) and therefore ignored when possible; some still draw it, e.g., the original CD4066 datasheet is a good illustration of CMOS, with proper symbols, and showing different substrate connections as well (necessary for the bilateral switch action).
Even worse, the alternative MOSFET symbol which looks more like a bipolar transistor reverses the direction of the triangle so it backwards from the direction of the body diode. I have never used that one but it does have a certain appeal since it looks consistent with bipolar transistors.
Yeah. This, I'm pretty sure, is a corruption, and best left forgotten, or preferably, burned in a firey pit.
Tim