Transfer from exchange to my Electrum cold wallet I have created a wallet on an offline computer and regenerated it as a watching-only wallet on my online computer using the public key of the offline computer wallet.

So far, so good. Now, how can I transfer my bitcoins from the exchange service to the cold wallet? Is it fine to just enter the first address I get when clicking the "receive" tab on the online computer (watching-only wallet), will bitcoins transfered to this address also 'arrive' in my cold storage (offline computer) wallet original?
The assumption stated at the end of your question is correct: 

You can send bitcoin to the address that is presented by Electrum's Receive tab and the funds will be sent to that address - accessible (spendable) from the cold wallet.

Electrum's Receive tab seems to choose a random address from your wallet's address pool. If you want to use a specific address, or a new address for every funds deposit, then go to the Addresses tab, right-click on a specific address and choose Request Payment from the drop-down menu.

Note: the xpub (extended) public key format is used by BIP32 deterministic wallets (such as Electrum seed wallets) to recreate all of the addresses associated with your xprv (master) private key. This master private key is stored in your cold wallet but not in the watch-only wallet that used the xpub key (at watch-wallet creation time) to generate the addresses it will watch.

By using the xpub key associated with the xprv key, your watch-only wallet contains all of the same addresses that were generated when you created the cold wallet. Anytime you manually generate a new address in the cold wallet you will have to manually import it into the watch-only wallet.

Never reveal your xpub key to anyone as they will be able to recreate all of your addresses and your privacy will be compromised.