If you're merely interested in the current "standard" accent of Japanese words, try one of these learning resources.
The jokes in this video are based on a more advanced topic of the Japanese pitch accent, namely アクセントの平板化 ("flattening"). Briefly, it refers to the change of the pronunciation of certain words from the non-flat to the flat (or "unaccented", "monotonous") type. Many common words underwent 平板化 in the past 100 years (e.g., グラス【HLL】 → グラス【LHH】).
For some words, both non-flat and flat versions are widely used today. A good example is 彼氏; both カレシ【HLL】 and カレシ【LHH】 are common, but the latter sounds younger and informal. Flattening usually does not change the meaning of the word, but occasionally a flat version gains a different nuance. For example, クラブ【HLL】 means generic "club", but クラブ【LHH】 often refers to nightclubs. ネット【HLL】 usually refers to netting and ネット【LHH】 usually refers to the Internet.
Generally speaking, when two types of pitch accent are in use, the flat version (indicated with
⤴ in your video) tends to sound younger, more slangy, jargon-like, or sometimes frivolous.
As for 梶さん, かじさん【HLLL】 is the orthodox pronunciation. Saying かじさん【LHHH】 is not particularly strange to my ears, but if someone does this intentionally, as in the video, it sounds like they are emphasizing its frivolous, slangy or jargon-like quality. For example, at 0:05 and 1:56, guys are saying 梶裕貴じゃね⤴ , and they are mimicking a チャラ男's way of speaking. At 2:20, Kaji himself is playing the role of formal 梶さん⤵ and frivolous 梶さん⤴.
Note that all the people in this video are professional seiyu who know how 平板化 works. Ordinary people are normally unaware of this.