As Jason says Clarkson screwed shank is now primarily for the legacy folk (I have a lifetime supply).
The mill – lathe compatible thing is something of a chimera. You don't need many milling cutter holders so changing everything just to suit the mill is something of a waste.
If I were starting out now I'd choose side lock holders and Weldon flat cutters. Weldon flat cutters are now readily available at homeworker friendly prices. If you luck into a stash of plain shanks the flat is easily fitted. Big advantage of side locks over ER is that they are shorter, essentially immune from pull out issues and don't need a hefty heave on the spanner to tighten them up. All good things when using on smaller machines where space and stiffness is limited.
There seems to be considerable legitimate debate on how tight an ER collet needs to be to hold well enough for our purposes. I suspect that very small differences in quality, accuracy and finish of both collet and cutter make a big difference to how tight they need to be for our use.
Buy the holders as you need them, you may be surprised how few you actually use. For my Bridgeport I have full sets of ER32 along with native R8 finger collets in both metric and imperial as well as the Clarkson that I use for most jobs. I also have the full range of side lock holders. Most of the collets I have never used in 20 years.
Frankly, with 20/20 hindsight, the money tied up in them would be better invested in extra side lock holders so I could leave some of my favourite cutters ready mounted CNC style. But I got nearly everything at very attractive prices so not too bad really.
Clive