That site says "BLB tube Peak: 352nm or 368nm" (dotted line) which means they don't know what they're talking about/ selling..
Just like the white looking tube the black looking tube has a phosphor with a peak emission at 352nm (and a range of 295-425nm).
Then for the "black" tube it says "Deep blue filter glass absorbs visible light". That's either Wood's glass or just an enamel coating, having a peak transmission around 365nm. Thus the BLB tube gets its peak output moved to 368nm (and a range of 328-408nm). It'll be "safer" to look at but will diminish the wavelengths of interest also.
PHILIPS TL8W/05 tube has its peak at 365-370nm (but ranges 300-460nm)
Datasheet/spectrum:
www.polychrome.nl/file_download/18/Philips-05.pdf
All these lamps will work but you want a(n unfiltered) peak as close to 380nm as possible for maximum efficiency, so the TL/05 seems better suited.
A 100W incandescent bulb will do the job too, it just takes a little longer.
Eye damage is like for a sunburn. You don't feel anything while it's going on. It takes a while to build a damaging dose and so any pain will come when it's already too late.
I believe we're talking minute exposure here rather than second.
I don't think that circuit will work either, it would require two separate coils & cores.
Connect the tubes like at the end of page 4 here:
http://www.capakor.com/.../(37)General_technical_information_of_fluorescent_lamp_starters.pdf
Use St 151 series starters.