Originally Posted by Supamax
Nope, I'm sure, and even the website you linked proves me right.
None of what you posted is incorrect, but you're missing one important detail: in interlaced modes, all the even lines are transmitted in one field, and all the odd lines are transmitted in the other field, and the two types of field have a different sync pattern in the VBlank, which allows the TV to distinguish them and offset the odd lines, achieving the interlacing.
The mode people are calling 240p or 288p is output by almost all video game systems from the 80s and early 90s, and involves either only odd fields, or only even fields. The sync patterns don't alternate each frame - hence no interlacing occurs. The same 240 / 288 scanlines are drawn each time, and the normally-interlaced scanlines are left blank - hence the black scanline emulation you find in certain emulators. So while the display timings are basically the same as 480i / 576i, these modes are not the same as an interlaced signal with successive 50Hz frames assigned to alternating interlaced fields.