Quibbling over techicalities isn't really going to help call it what you will disputed bill disputed account, aside from their dictionary terms it all boils down to the same thing for this individual.
Looking through the infomation presented I would make the assumption that the account has been in the landlord's responsibility for some time, and the landlord has done very little until recently. This would account for the OP not recieving a bill until recently. There are a number of reasons this could be. The landlord could simply not have been bothered, he could have had a letting agency who should have been managing the account or several other things. However it's come about the landlord has presented a tenancy agreement clearly showing periods of responsibility. If my assumption is correct sadly you cannot blame the supplier for this.
I'm having to agree with the representative from e.on it most defiantley is the responsibility of the new teant or new occupier to inform suppliers they moved in, as last I checked the crystal ball is nothing more than a childs toy and as yet not every property is wired up to a coverty surveilance system.
In an ideal world the landlord would provide or letting agency will provide the teant a list of who supplies what with contact details, again falling back to my earlier assumption it would be a relatively safe bet that the landlord/letting agency didn't do this.
When British Gas took over any supply at the property they would have sent you a sales infomation pack. This contains your details, the plan you have signed up for on what fuels, details fo your payment method and the terms and conditions that apply to you. I'm assuming you didn't read this through thoroughly, as this would have been a key indicator as to what is going on. If as you say British Gas never applied for the second fuel there would be no contact with the "losing supplier".
Now moving on to something a little more constructive. On the value of the infomation you've presented. E.on have rightfully supplied you the energy you have consumed and that energy needs to be paid for, there are no grounds for them to consider reducing the balance so it must be paid in full. i would assume however that givne the circumstaces they would be willing to set up a suiable repayment plan. If I were in your shoes i would now be speaking with British Gas to find out why they did not act on your intial instruction, you may be able to secure financial compensation to help with your e.on balance. In addition I would also be speaking with my landlord and finding out why it's taken him nearly 4 years to inform the supplier he wasn't responsible.
This is all hinged on weather my assumption regarding an account being active for your landlord. you should consider requesting a SAR (Subject Access Request) from e.on to confirm or disprove this assumption. You will have to pay £10.00 and the request cna take up to 40 days to be completed. You will be presented with a copy of every piece of infoamtion e.on has about you, recieved from you, or on your behalf as well as what they have sent. I beleive they may also have date and time stamps where relevant to show a clear time-line for you.
If there is not a reference to any activity prior to the 10th of June 2009 then your supplier is absolutely correct in the situation and my assumption would appear accurate, if this is not the case then at east we'll have more infomation to build a case on.
Whatever you decide to do let us know how you get on.