By cheaper transactions do you mean on the blockchain network?
The fee you pay for sending bitcoin from an uncompressed address is slightly smaller than the fee you pay for sending transaction from a compressed address.
Therefore a legacy address is uncompressed and a SegWit address is compressed. Correct?
First note that when we say "uncompressed address", we mean that the address has been derived from an uncompressed public key and when we say "compressed address", we mean that the address has been derived from a compressed public key.
You can generate two different legacy addresses using compressed and uncompressed formats of your public key. They are both standard and you can use each of them, however it's recommended to use the compressed address. Most wallets supports only compressed addresses.
When it comes to bech32 addresses, you can still generate a one using each of compressed and uncompressed formats your public key, but the one that is generated from the uncompressed public key would be non-standard and it would be very difficult to spend fund from that.