So with this online service the wallet file is kept on my HD while the transactions are initiated via the website, right ?
If you are referring to using an online eWallet service like MyBitcoin, then when doing so you have simply just an account on that service. There is no local software and no local wallet related to that eWallet service.
It is just like online banking with your bank. Except ... of course, MyBitcoin is not a bank, it is an eWallet provider.
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http://en.bitcoin.it/wiki/EWalletAnd I'm still not quite clear about this, is there one wallet file per user or one wallet file per installation (of the client or per account on this website) ?
When installing and using the bitcoin client it will, by default, expect the wallet to exist in a specific directory location. If the wallet does not yet exist one will be created.
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http://en.bitcoin.it/wiki/Data_directoryAs far as why your download was taking a long time, make sure to use the newly released version of the Bitcoin client, v0.3.23. If has improvements specifically to help with the problem you are describing:
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http://www.bitcoinnews.com/post/6521309281