If using a mag base for flexible mounting put a small, thinnish, plastic bag round the mag base and tie the neck to intercept any ferrous swarf. Periodically remove the whole thing and peel the bag off with the magnet off to remove the swarf encrustation without tears. With care you can even re-use the bag a few times. The thicker varieties of cling film also work but one time only and need care in removal because swarf can cut through and stay behind. Naturally its the really sharp bit!
Various breeds of inexpensive flexi stalk desk lamps that will work well when transplanted from the weighted foot to the mag base. The IKEA versions are said to be good although not the cheapest. They do one type with a big bulldog clip type attachment which might do well on the splashguard. I got a bunch from LiDL but have yet to use them. In retrospect a battery powered lamp would be better. No trailing wire to worry about.
My main lights are folding rigid arm types with Edison Screw LED bulbs fitted after changing the holders. The one on the Smart & Brown, a relatively short machine, is mounted on the wall above and behind the tailstock. The P&W is longer so its fixed to the middle of the spashguard. Both give decent coverage of the work area. Professional standard lights originally sourced from RS (but at surplus prices) so I'm unworried by 240 volt mains power. Bases are fixed anyway and I don't use the built in switches.
I also have area lights under the shelf above.
Much to be said for wiring the light direct to the main lathe power switch to act as a power on indicator. When the lights on the lathe can run. If nowt else it will remind you to switch the thing off when you go in for supper.
Clive