in- ‘upon’ + sistere ‘stand’

source: [日常の笹原幸治郎が真剣に主張しているGIF画像｜無料GIF画像検索 GIFMAGAZINE 500556]

demand something forcefully, not accepting refusal:

〈人が〉 «…を» (強く, しつこく)主張する, 言い張る «on/upon (doing)»

e.g. she insisted on carrying her own bag

e.g. I'll call him and cancel it, if you insist.

(insist on) demand forcefully to have something:

«…を» (強く)要求する, せがむ «on/upon (doing)»

e.g. he insisted on answers to his allegations.

(insist on) persist in doing something even though it is annoying or odd:

〖～ on/upon doing〗 (迷惑にもかかわらず)…し続ける, …するのにこだわる

e.g. the heavy studded boots she insisted on wearing.

reporting verb maintain or put forward a statement positively and assertively:

〖insist (that)節〗 (疑念･反対に対して)…だと強く,しつこく主張する, 言い張る, 強調する

e.g. with clause : the chairman insisted that all was not doom and gloom

｟書｠ 〖直接話法〗…だと強く言う

e.g. with direct speech : “I really am all right now,” Isabel insisted.

late 16th century (in the sense ‘persist, persevere’): from Latin insistere ‘persist’, from in- ‘upon’ + sistere ‘stand’.