OpenSSH is the de facto standard implementation of the SSH protocol. If PuTTY and OpenSSH differ, PuTTY is the one that's incompatible.
If you generate a key with OpenSSH using
ssh-keygen with the default options, it will work with virtually every server out there. A server that doesn't accept such a key would be antique, using a different implementation of SSH, or configured in a weird restrictive way. Keys of a non-default type may not be supported on some servers, in particular ECDSA keys make session establishment very slightly faster but are only supported by recent versions of OpenSSH.
PuTTY uses a different key file format. It comes with tools to convert between its own
.ppk format and the format of OpenSSH.
This ssh-126.96.36.199 you found is a commercial product which has its own different private key format. There's no reason to use it instead of OpenSSH, it can only be less compatible, it requires paying, and there's about zero tutorial on how to use it out there.