I am looking for something that would allow me to offer a general "deposit" address (like for payments through my website, services I offer etc.).
I am using Electrum as cold storage solution (with a watch-only copy created on an online computer). Can I just create 1 'Receiving address' with Electrum and display this address on my website for payments of any kind, or would I have to create an address for every indivicual payment? I just don't want anything to go wrong...
As mentioned in a comment to your question, reuse of addresses is discouraged. It's considered good practice to generate a new address for each receive transaction.
This question has a detailed answer that discusses how Electrum generates its initial keypool, and how generating additional addresses should be handled.
As outlined in the Electrum docs you can generate a new address in the Electrum console with:
wallet.create_new_address(False)
Be aware of the gap limit issue (discussed in the same document).
If you would like to automate the process you could write your own script or use one of the many open source utilities available on GitHub to generate a new private key and associated address. Let your web page display this unique address for each new transaction. Later, import this address's private key to a non-seed Electrum wallet. Of course, a seed (HD) wallet will not accept addresses that were not specifically generated by it.
Alternatively, and for greater security and control, you could run an instance of Bitcoin Core. It can provide your web server with a unique address via an RPC call. There would then be no need to manually request a new address or to later import the private key.
