As a Brit who uses far more stranded wire than solid [I don't do first fixes, that's someone else's job], there are two ways to do it 'properly', but this can depend on how much room you have in your screw-down terminal.
Proper-proper. Big terminals & junction boxes.
Strip 15-20mm of the sheath; twist, fold in half, then insert & screw down.
What actually happens. Small terminals & household plugs.
Strip 8-10mm of sheath; twist, insert & screw down.
In each case there should be little to no 'spare' hanging out the far side of the terminal & no exposed wire on the nearside.
Stranded doesn't look like solid in a terminal after screw-down. It compresses & the exposed end flares - which is why you keep it short in small terminals. No spare, no flare.
Ideally, all your exposed portion fits exactly inside the terminal.
Pre-made cables often have copper/brass ferrules crimped onto the cable. Once you're cutting your own, you can't do this. I've seen cables soldered first - but having been in the position of re-doing a lot of these several years later, I'd say there is a fair chance the wire gets part-cut & compromised by doing this; I've nothing more than "I don't like it, it feels dodgy".
Bare wire 'just works' so long as you're not trying to share the terminal with more solid core than stranded. Stranded bunches up nicely & stays firm. Solid core can upset that if you have for instance 3 solids to 1 stranded in a big terminal; the stranded can be free to move in the gaps.
The recognised test is to see if you can move it once it's tightened down. [Empirical science FTW ;))
If not, you're good to go.
A note on twisting.
Some people like to leave the end of the sheath on, so they can quickly twist without getting spiked by spare bits of strand. I've never bothered. It takes too much time for very little difference in the end result. As I use an automatic stripper, this would just be far too much faff to bother with.
Plus, if your wire end is scrunched up enough to spike you as you twist it, you should have started with a clean end, not re-use the last one someone else made.
Notes prompted by answers elsewhere.
We don't use j-bends on terminals in the UK, ours are all straight-through. I've only ever seen them on hifi speaker terminals, where they're more faff with potential for inaccurate clamping than I'd ever like to see in a pro wiring job. There was a brief penchant for the j-type in some domestic plugs a decade ago - thankfully they seem to be gone now.
It seems that some territories in the EU mandate DIY slip-over ferrules which are then squeezed down before inserting into the terminal. I really can't comment on those, never seen one in action.
If you're going to be doing this a lot, I'd recommend a semi-automatic stripper. Set length, insert wire, squeeze. Perfect size every time, whatever the wire thickness.
Not one of this type - they need adjusting for thickness every single time. PITA.
[Had to use one of these as I didn't have my own kit with me to swap a slew of plugs at the in-laws last weekend, after they moved back to the UK from Spain… there was swearing after the 4th appliance with yet another different wire gauge;)
Something more like this [this isn't one I've ever used, but I can't find a pic of mine] Doesn't matter how large the wire diameter is, it 'just works'.
Tricks of the trade… [or old man's idle musings]
I used to some days swap 50 - 80 13A plugs a day [I worked in retail maintenance; multimedia, displays & small electricals, often converting EU imports to UK spec, on site, no workbench.]
I had a plug swap down to under two minutes ;)
Clip old plug off, strip outer, cut wires to three different lengths, strip ends & assemble. I worked "on a price" [paid by the task, not by the hour], but someone would QC us at random, so these all had to be good.
If you get a stripper, get one that will cut the whole wire first. Then your toolkit for this is a drill-driver, 'Stanley' [modelling knife, US?] & the stripper.
Check this gap, which must be big enough to cut the entire cable… cheap ones can't, good ones can.