What to listen to
You should be listening for changes in the fundamental frequency:
The fundamental frequency, often referred to simply as the fundamental, is defined as the lowest frequency of a periodic waveform.
While formants (aka harmonics), also reflect pitch accent, they may be harder to follow due to fluctuations from vowel shape, and certainly sometimes harder to identify in a spectrogram.
How to best use the site
As for use of the linked site, I recommend first limiting it to heiban and atamadaka only, until you can determine the difference between those two. Maybe try some basic musical exercises in identifying lower vs higher pitched notes if you’re having issues.
How the types map to actual pitch
Also, since it seems like you are maybe having some issues with understanding the types:
定か サ＼ダカ ＨＬＬ atamadaka
斜め ナナ＼メ ＬＨＬ nakadaka
車 クルマ￣ ＬＨＨ heiban
頭 アタマ＼ ＬＨＨ odaka (note this is exactly the same as heiban, the difference only becomes apparent when something follows the word like a particle)
As you can see, the first mora starts low, then following ones are high, until the downstep, and then stuff after that is low.
However, if the second mora is ン or ー, then the word will start high:
英語 エーゴ￣ ＨＨＨ heiban
今後 コンゴ￣ ＨＨＨ heiban
Also, especially in the audio used on that site, heiban words said in isolation will have a gradual drop throughout them, which is hard to represent with just L and H, but you must learn to differentiate them from accents via sufficient exposure. You could for example notate the audio for 英語 and 今後 on that site as ＨＨＭ if you wanted.