2. I'd prefer to keep the camera off as I'm just relaxing today
3. You wouldn't want to see me - trust me!
Well, there is no obligation for you to give an explanation to your teacher. Having the camera turned off is a valid option and it's taken up by great numbers of students every day. So phrase 1 is fine. However, if you prefer to give a reason, better would be to say you are just relaxing, rather than embarrassed. That is quite enough explanation. If you feel like a making a joke of it, phrase 3, may be suitable.
I a little bit shy so I would prefer that I keep my camera off.
If someone is shy it means that they are nervous and uncomfortable with other people.In this case you may be too shy to speak to the teacher in English and you can gain some confidence if you speak while your camera is off.
For example:Children are often shy of people they don't know.
誰かがshyであると言ったら、それは彼らが他のひとといるときに緊張したり、居心地が悪いということを意味します。
この場合では、恥ずかしすぎて英語で先生に話しかけられず、カメラがオフになっている間は少し自信を持って話せるということでしょう。
例：
Children are often shy of people they don't know.
子供はしばしば知らないひとに対して恥ずかしがる。
I would rather have/keep my camera off as i'm embarassed
I am a little bit shy so would like to keep my camera off
If you don't mind I would rather not have my camera on as I'm shy/embarassed
If you are asked whether you want to have the camera on then you could say 'I would rather have/keep my camera off' or 'If you don't mind I would rather have my camera off' by adding I'm a little shy/embarassed you are explaining why
カメラをオンにしなくていいのか尋ねられたら、次のように言えます：
'I would rather have/keep my camera off'（カメラはオフにしたいです）
'If you don't mind I would rather have my camera off'（もし良ければ、カメラはオフにしたいです）
また、'I'm a little shy/embarassed'（恥ずかしいので）と加えると、理由を伝えられます。
Native speakers will say like this: “I'd like to keep the camera off, as I'm shy”.
There are lots of other ways to say it, but this one is the most accurate and most commonly used.
Here is an example on how to use this in a conversation:
A: You can turn on your camera if you want to.
B: I'd like to keep the camera off. I'm quite shy.
ネイティブスピーカーなら次のように言います。
“I'd like to keep the camera off, as I'm shy”.
（シャイなので、カメラはオフにしておきたいです）
他にもいろいろな言い方ができますが、これが最も正確で一般的です。
以下、会話例です。
A: You can turn on your camera if you want to.（カメラ、オンにしたかったらしてもいいですよ）
B: I'd like to keep the camera off. I'm quite shy.（カメラはオフにしておきたいです。シャイなんです）
"Sorry, I would rather not turn the camera on today"
"I would rather leave the camera off. I can be quite shy".
In order to politely explain that you would personally prefer to keep the camera off due to embarrassment, you could state "I would rather leave the camera off. I can be quite shy". You could also politely say this without mentioning your embarrassment by stating "Sorry, I would rather not turn the camera on today".
恥ずかしいのでカメラをオフにしたいと丁寧に伝えたいなら、次のように言えます。
"I would rather leave the camera off. I can be quite shy".
（カメラはオフにしておきたいです、ちょっとシャイなので）
また、次のように、恥ずかしいということには言及せず伝えることもできます。
"Sorry, I would rather not turn the camera on today".
（すみません、今日はカメラをオフにしておきたいです）
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