the word warranty usually does not mean much, it needs proper definition of what is guaranteed and what's not, the use and the length of time to assess what it's worth.
I'm sorry but this just isn't true. At all. Hillas v Arcos. Honestly if you knew how wrong you were you wouldn't even begin to post on these topics. I'm holding back from being blunt because I know how awful I find being told I'm talking nonsense. But I'm afraid you are.
The courts will look to the intentions of the parties. A seemingly vague word can and will be construed by the court if it's thought capable of being construed, and if the parties were thought to have intended to create legal relations. In this case the natural meaning of a warranty, is a promise that the item will function properly in the absence of abuse. If the item fails, what happens next is a question of what is fair. For a powerful company it seems fair that it will be made to work again or similar. For a private seller I think a refund less a pro-rata deduction for usage. These are easily capable of being ascertained.
Warranty is closer to an insurance policy than the sales of goods act.
Your understanding of warranty from my having read your posts on another thread which I gave up on in despair, is unfortunately limited to your having just read warranties, without your reflecting on the application of the principles of contract law to these contracts of warranty.
The component is not fitted by the vendor or one of his agents, the battery is not supplied by the vendor or his agents, the warranty terms are not properly defined to be part of a sale contract. If the OP paid with paypal then he may get some help.
If the vendor is an individual, having bought the kit from China, surplus to his requirements, tries to sell it as seen (the default term of items for sale on free ads) on gumtree and the OP paid in cash, I don't think he'd win in Small Claims Court.
The issue of being a business seller aside (which is a simple question of fact imho, he is or isn't), all we are talking about is a promise of a warranty. Paypal may not be capable of incorporating unwritten (spoken only) terms into their guarantee, although this I don't know. If it's written then it's possible PayPal could be held to honour it although this depends on an assessment of whether PayPal's contract encompasses that on the facts, and whether any exclusion of PayPals is incorporated and if so whether it meets fairness criteria.
However the small claims court is the other possible route if any promise of warranty had been made by whatever means. Whether someone promises something is a question of fact not of law, and if it is a concluding part of a negotiation for a sale then it is ordinarily incorporated into the contract. The court will just ask whether they believe the buyer, which on the facts I think they might. If they believe the buyer that a promise was made, they will have no choice but to enforce the contract.
I am sorry but I simply don't have the time and energy to keep correcting you. I will say it - you talk nonsense on contract. Please spare me the horror of your spreading utter misinformation about the law. If you're really interested in it take a course in contract and consumer, or read some serious books - but please, please stop pumping out this utter nonsense. I am sorry for being rude, genuinely I am. I really look up to your knowledge on all sorts of other things but not consumer law.