I am not sure, but this is what blockchain explorer writes about this transaction:
You say that you checked the transaction on the blockchain explorer, but in fact you checked the balance of the address. You need the TXID to find the transaction on the blockchain explorer. If your transaction was confirmed on the network, you should have its TXID.
Your seed phrase is all you need. If you import your seed phrase in a new wallet and select the correct script type and derivation path, you should see the same addresses.
No I did nothing about script type and derivation path. So in the lack of these settings, is there no way the BTC arrives and the verification can work?
What do you mean by that? When you import your seed into a new wallet, it asks you which script type, i.e. which type of wallet you want. If you choose the wrong one, you will get a completely different set of addresses in your wallet.
But instead of testing with transactions, and potentially losing money, why not simply compare the addresses in your "hot" wallet and your "cold" wallet to see if they match?