For what it's worth I started aged 7 with a 3/4 size and this possibly led me to very strong left hand technique, I'm unsure why this happened (probably a deal on the guitar via my teacher). I always appreciated having a nice instrument even if just to gaze into the wood sometimes. I had an innate musical interest but perhaps this did also somewhat influence the attention I paid to it, hard to say.
My fingers are literally longer on my left hand which arguably lends to so-called "prodigious" results. I'd err on the side of quality sub £100 but that's me. You can start and play well on any guitar that is set up and has new strings, and I often enjoy playing on borderline broken guitars as a challenge.
I'd definitely start with a classical style guitar. Steel or even electric are far to tough on young skin and the latter I would say actually takes more technique to sound like you would want it to sound. The former naturally leads well into the latter because of promoting some of the most essential techniques and being slightly harder to form undesirable habits on.
Starting young is a very good idea and can equip someone with a both a mental and creative tool for neurodevelopment especially in regards to dexterity, regulating and expressing emotion and essentially a sort of physical arithmetic, and source of soul and connection for life. It's a lovely thing and I wish you all the best of enjoyment with the process.
Edit: I just looked at the photo on my windowsil when I was maybe 8, 9 or 10 and spotting the J in the sound hole, that would appear I had a Joan Cashimira then, too. I don't remember this because I was too young, but I still have my fullsize about with a slightly smaller body than normal (maybe a quasi tenor guitar), that I picked myself from the guitar luthier who used to live next door to me. And that instrument still sounds fantastic. Something like this.
Those are midrange guitars, even at that size, probably around the £200-£250 point. My mother is/was an amazingly generous person, and we actually weren't wealthy. This surprises me, as this wasn't long after my father "moved on" leaving us with little to no money. My teacher was essentially a surrogate in some respects.
I guess I can only say, I appreciate that then and appreciate it now, too. But I still play average, random knockabout guitars in flea markets etc and think they sound objectively musical, so I'm not sure how important that actually is - especially at such an age. I suppose treasures are important and it was that; I only sold it because it was too small and I needed to fuller sounding, full scale guitar to move to grades 5 on up to, and I was told I had to do that to pay for a new guitar.
I do remember selling it for maybe £80 to £120 to get another instrument, so perhaps it's with bearing in mind good quality small guitars are in demand. To write off £20-40 quid may or may not be worth the "trial period".
I think having a fantastic teacher, being shy and somewhat isolated and so having something to "communicate with" (and to) had more to do with it than just what it was worth though. But I could tell it had intrinsic value and sounded good, which spoke to me.
I would avoid anything that is painted or has transfers on it. If it's not wood or that element is being hidden, it's going to sound flat. Definitely not composite materials (MDF, plywood etc).
