Helmets used for sprints and hillclimbs need to be BS 6658-85A (what was blue sticker); you do not need the fire resistant version (red sticker), nor do you need to fork out for a gold sticker lid… Trackdays vary; I’ve seen some trackday organisers accept green-stckered moped helmets, or even, on one occasion, a police riot helmet. If for competition (sprints, hillclimbs), it will also need to be scruitineered; any damage to the helmet might mean you will fail scrutineering. Helmets for wheel to wheel racing will need to be of the first resistant lining type.
Gold stickered helmets will generally have a nicer finish and fit, compared to cheaper ones. In principle, they have been subjected to additional QC procedures (ie. percentile fewer fails).
Helmets vary considerably in fit and comfort; different manufacturer’s sizings are often very different. Given that you can pick up a perfectly good blue stickered helmet from a motorbike shop for £50-60, if you are intending on doing more than one trackday, it doesn’t make a lot of sense renting one (which will cost you £20+ by the time you’ve factored in postage), particularly as you can spend a bit of time sorting out a helmet that actually fits your head, and that isn’t too uncomfortable for 10-15 minute stretches. You don’t want a helmet that rattles around on your head, loose.
Full face or open face. Some events may actually ban open face helmets in open cars, because of the risk of debris taking out your teeth. There aren’t that many open faced, blue stickered (or the newer equivalents) helmets at the cheaper end (OMP and Sparco clubman lids are £120ish) anyhow.
Given the other expenses required for participating in bone fide motorsport events, it seems a bit odd why you would want to rent a helmet out every time. Average sprint/hillclimb season; £50-100 per championship, £50-100 per event (say 6 or 7 events per summer season), £100 for a cheap Proban suit, £50ish for a non-race natB licence, £50 for miscellaneous bits and bobs (numbers, light breaker etc); best part of £1000 spent. Spend £140 on a rented helmet per season, which doesn’t really fit, and will need scruitneering each time (first time; you get a MSA sticker, which they will charge you £1 for, to save time on subsequent events that season): so actual cost is more like £147.
Loan lids are good for those who have never tried a trackday before, or are looking for a spare for a passenger, but its much better/nicer to have your own. I feel the same about bowling shoes…