"AC97" and "HD Audio" reffer to Intel standards for onboard audio. A minor part of those standards is a front-panel audio connector.
However the connectors are only semi-compatible. In particular the handling of plugging in of front panel devices is different.
On AC97 audio to the rear audio output was looped by the front audio output and plugging in a device to the front panel audio output would electrically disconnect the rear audio output from the audio source. If you wanted to use an AC97 motherboard without front panel audio it was nessacery to put jumpers on the audio header to make the rear audio output work.
On HDA audio to the front and rear outputs comes from seperate outputs on the audio codec. The pins that were previously used to loop back audio from the front are repurposed as connector-detect pins. It is up to software whether to disable the rear panel audio output when a connector is plugged into the front panel output.
So if you plug a HDA front panel into an AC97 motherboard you will get no output on the rear audio output. Most users aren't going to find that acceptable.
If you plug an AC97 front panel into a HDA motherboard then plugin detection will not work correctly, it may detect devices as not present when they are present and vice-versa and the exact results may depend on what audio is playing. On some motherboards it may be possible to disable the plug-in detection so they can work better (though still not perfectly) with an AC97 front panel.
Your particular front panel looks to me like it is primerally designed for HD audio motherboards but with a hack to give limited support for AC97 boards. There are loops of wire on the AC97 connectors which will make the rear output work but it won't be switched by the front panel connectors like it would be with a proper AC97 front panel.
Intel introduced HDA back in 2004 though both AC97 and HDA were supported in paralell for a time but anything even remotely modern should be HDA. According to it's datasheet the ALC888 is a HDA chip.
I would suggest trying the HD audio connector first, if the rear audio output on your motherboard doesn't work try using the AC97 connector instead.