I remember your previous posts (especially your horse)!
I don't think it's a failure at all to admit to yourself that something isn't for you. In my past I have made "mistakes" that I felt very guilty/ ashamed of at the time - e.g., big life commitments that then weren't working out - but in retrospect actually feel grateful for the learning experience and proud of myself for being able to recognise when something wasn't for me, and being able to back out quickly instead of persevering due to misguided feelings of shame/ guilt/ obligation/ duty.
From the sounds of it, you are very much a "people person" - you need relationships, friends around you, company, an intimate relationship. Egged on by others, you decided to make the move to London, perhaps without realising what that would actually entail and being caught up more in the fantasy of living in London, rather than considering the practical reality. Only in moving to London have you realised what you've lost, which you may previously have taken for granted - i.e., your loved ones and your established support network. Now you are worried about "losing face" by returning quite quickly to your home town.
I think it would help you to write down the things that make you happy/ fulfilled. Is it close relationships, time with your horse, or partying/ art galleries, etc.? Then consider which of these "happiness needs" you can meet living at home versus living in London. You can meet certain needs without having to live in the location where they are most prevalent (e.g., meet the need for attending art galleries through holidays/ day trips, rather than living amongst art galleries). It may also help to complete a 2x2 table listing the pros and cons of each decision - "moving back home" versus "staying in London". Look at where the most meaningful pros/ cons lie, and evaluate whether for one decision there may be a huge list of pros and not that many cons, for example.
Ultimately I think it's important for you to learn (if you haven't already) that happiness is not about your environment/ circumstances - it is your personal, internal responsibility. In order to be happier I don't think you can look to your surroundings (though they help) - I think you need to focus on things like developing yourself as a person, learning to be comfortable with who you are when others are not around, identifying your interests/ skills and developing these, trying to address any personality weaknesses, etc. But I think good relationships are also very important for contentment/ fulfillment. There is nothing wrong with you and you are not "one of those people who is destined never to be happy" - there is no "destiny" here, only personal responsibility (in my opinion), but you shouldn't feel bad about who you are or what choices you need to make in order to feel most happy. Don't be seduced by "the London dream" or whatever if it's just not for you. It's not for me, for example - I value my relationships too much and will likely always live in my small-ish home town because my loved ones are most important.
I know a girl from school who always attributed her unhappiness to her surroundings ("once I leave school I'll be happy", "once I change uni courses I'll be happy", "once I change careers I'll be happy", etc.). She's made a lot of changes and she's still not happy. Changing your environment doesn't change your mental state/ way of thinking/ mental health/ happiness - only you can change that, through hard work!