Can't tell you about the car, and like some have said the connector on the dashboard is confusing. The answer from the company suggests the charger was external and plugs into the connector. The look of the connector and apparent lack of physical polarisation suggests mains to me. You need to investigate that.
However. I can tell you what the battery is and exactly how to charge it.
It's a 12volt sealed lead acid battery, or lead acid valve regulated battery, as they like to call them now. The charger it requires needs to suit the chemistry of the battery. A proper lead acid battery charger has a 3 stage charge. 1) a bulk constant current charge, followed by 2)a constant voltage charge, followed by 3) a constant voltage float charge. To protect the battery and give its best life the charger needs to do this.
Do a net and ebay search for Yuasa YCP2A12 12v 2Amp charger or the slightly older CM1.5A12T 12v 1.5Amp model. You're looking at about 30 quid + P&P. Your battery has a 10Ah capacity so the usual max charge current would be limited to 1/4 of its capacity. So 2.5Amps. The 2Amp charger therefore will be fine. The 1.5Amp model will also be fine just a little slower at doing its stuff.
You gain a number of things by getting a proper charger for it.
1) The charger will correctly charge the battery and look after it at the end of the charging cycle. A red light tells you its charging. A green light will light to say its done. Often combinations of the two will report other useful information as per the manual. No guesswork involved.
2) You don't need to worry about it whilst its charging or make sure you remember to take it off charge at the end, the charger will go in 'float mode' and it can stay like that indefinitely without damage until you decide to use it. Cheap chargers, if you forget about them, will harm the battery.
3) When the car doesn't work anymore the charger will be handy for charging any 12v lead acid battery over 8Ah capacity. Although it will be slow doing it it will happily charge car batteries and will do so in a kinder way than the majority of halfords style car battery chargers. Especially when the charging cycle ends and you've forgotten about it.
4) Charger will be short circuit and reverse polarity protected.
The 12v 1000mA one on ebay I wouldn't bother with. There are loads of so called chargers out there that do a half baked job. That looks like one of them. If its 12v out it wont charge a 12volt battery fully either, or very quickly.
If you can take the battery out of the car to take its photo you could charge it like that, standalone on the bench. Charging it in the car would expose the car circuitry (if it has any?)to its max charging voltage, in this case about 15volts. I wouldn't of though it would be a problem but again you'd need to check.
Be aware that lead acid batteries have the ability to provide massive current / power that completely belies their size. So be sure not to accidentally short out its terminals.
Also, always store lead acid batteries in a charged state.
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