様(よう)に～て欲(ほ)しいWant ~ to ~ like ~, Want ~ to be able to ~, Want ~ to ~ in the way of ~

Both ように～てほしい and てほしい express wanting someone else to do something, with the key difference being the presence of よう. よう adds a nuance of doing something in a certain way. For example, 忘れないでほしい ('I want you not to forget.') vs. 忘れないようにしてほしい ('I want you to make sure to not forget.', in the sense of doing what needs to be done to ensure you do not forget).

ようにする means 'to make an effort to ensure something happens a certain way' and expresses intention or habit. For example, 遅れないようにする ('I'll try not to be late.'). In contrast, ように～てほしい expresses a desire for someone else to behave a certain way. For example, 遅れないようにしてほしい ('I want you to make an effort not to be late.').

ようにいう means 'to tell or ask someone to do something.' It is often used for indirect or general requests. For example, 来るようにいう 'tell (someone) to come'. When paired with てほしい, it shows a desire for someone else to make the request. For example, 来るようにいってほしい ('I want you to tell him/her/them to come.').

These two patterns using ほしい to express a 'want' differ in meaning and usage. がほしい follows nouns to express the speaker's desire for something, whether physical (金がほしい = 'I want money.') or figurative (自由がほしい = 'I want freedom.'). ようにしてほしい combines ようにする and てほしい to express a desire for someone to act in a certain way, e.g. 忘れないようにしてほしい ('I want you to make sure not to forget.').

ように・ような follows the plain form of a verb or adjective and expresses general similarity. For example, 日本人のように話す 'speak like a Japanese person'. The same よう appears in ように～てほしい, which expresses wanting someone to act a certain way. For example, 日本人のように話してほしい ('I want you to speak like a Japanese person.').

The core meaning of ように is 'so that' or 'in order to', and it is commonly used in patterns like ようにする 'make an effort to (A)' or ようにいう 'ask someone to (A)'. When used together with the ～てほしい form, it expresses wanting someone else to do the action. For example, 忘れないようにする ('I'll try not to forget.') vs. 忘れないようにしてほしい ('I want you to try not to forget.').