Edit: This is almost certainly a scam - I'm looking at the source right now, and they have replaced the actual Bitcoin seeds with IPs, so it is likely a cloned network that tries to trick you into accepting worthless coins. It may also steal your keys, but I don't have evidence for that yet.

Within Bitcoin, there is no form of private transactions (in the form that you will not see it on Blockchain.info).

Moreover, the website you linked to is almost certainly a scam - The footer claims that it is built by the Bitcoin Core Developers, which seems unlikely (archive.org). You can find the actual wallet/node built by them directly on Github -https://github.com/bitcoin/bitcoin.

It seems likely that they have either:

1. Created an altcoin network, and are distributing their own wallet that connects to their network while branding it as a Bitcoin wallet. This would trick you into believing you have received Bitcoin as payment, while you have actually received worthless coins from a cloned network. This would also explain why blockchain.info is not showing the transaction.
2. Distributed a malicious wallet client that is able to steal your keys in some way (or provide you with keys that are already compromised), so that if you receive actual Bitcoin, they can take it from you.

If your client wishes to pay in BTC, there are many safe ways to accept it. You simply need to download an actual, reliable bitcoin wallet, such as from the link above, or Electrum, or buy a hardware wallet such as a ledger or trezor.

Then simply provide the client with an address from your trusted wallet - As long as you receive BTC with multiple confirmations, you can be sure that it is valid. When dealing with someone who is already attempting to scam you, make sure you check multiple sources before considering the payment completed, such as checking your transaction on multiple, independent explorers, in addition to what your wallet shows you.