Yes, both are examples of the Noble False Widow, Steatoda nobilis, the species principally implicated in last years rather melodramatic media reports. This is not a 'deadly' spider by any stretch of the imagination.
The example that you say bit you is a mature male example and these leave their webs in search of 'lady Steatoda nobilis' . In their rather random wandering they frequently enter open windows and find themselves in odd places, such as within bedding or clothing. When trapped against our skin they will bite, though this is generally more mild than a bee or wasp sting though the likelihood of being bitten by a spider is still far less.
Treatment is simply to apply antihistamine cream if required and to keep the area clean and hence free from potential bacterial infections, particularly if it becomes itchy and you are tempted to scratch.
My rule of thumb is that I am happy with these living in my garden or on the outside if the house, any found setting up home indoors, or wandering males, are tipped outside.
Could you tell me where you have seen this by forwarding a grid reference or post code to me at email@example.com as it is valuable to know where these are being found so we can get a better picture of the current UK distribution of this species.