要：to want , 得 ： have to eg: 我要回家了。（I want to go home. “ personal wishes”） 我得回家了。 （I have to go home. maybe you dont want to go home , but it is too late or something ..）

You are right. When expressing obligation, they have similar meanings, while "要" is more formal than"得". We normally don't use "得", say, in official documents. Here are some examples: "中国共产党要始终代表中国最广大人民的根本利益" "要加强对权力运行的制约和监督，把权力关进制度的笼子里" If "要" is replaced with"得" in the two sentences above, they will seem informal and inappropriate. 我们得单独谈谈。 As in this sentence, if we replace"得" with "要", it may sound a little bit stiff. By the way, when we say that someone has to do something for their own good, we tend to use "得" (but not always). For example: 你得按时吃药，才能早日康复。 你得找个伴了。 In the two sentences above, the speakers are providing well-meant, sincere advice to the listeners, and by using"得" they can enhance that feeling of friendliness.

I understand what you say but I think that explanation is only valid when you talk about yourself, for example, when you say "I need to rest a little" (I want to rest a little / I am going to rest a little). But when you use 要 to say for example "before eating you must wash your hands" there is no "wanting" here, no "willing". It is just a moral duty or an obligation.

"要"shows more personal willings, equal to "want"in English, it means you have the desire/wish to do things. “得”means you have to do something because of the external circumstances or someone push you, not your innner desire.

我要走了=我想走/我正在准备走/。I want to leave this place or I am about to leave or 'beybey'. 我得走了=我必须走/某种情况或某件事迫使我离开。I have to go,I'm in trouble,My dad will beat me,I have a flight to catch,,,always something there push me to leave,or just a good explanation to leave the place you don't want to stick around,you pretend you have to leave. and sometimes they both can start from your inner motivation, please just try to fit them in different conditions to figure it out,