Four things which you need to know about the construction "had + better + V1":
1. 'Would better + V1' does not exist in modern English.
The only correct form is 'had better'.
In the phrase 'they'd better see him now', the 'd' is a contraction of 'had'. It can never be 'would'.
2. You said "I do think 'would better' stands for to prefer".
No, you are not correct. There are two mistakes here:
- 'Would better' does not exist.
- 'd better' does not mean 'prefer'. See #3.
3. A common mistake among non-natives, particularly Russian speakers, is to try to use 'had better' to mean 'prefer'.
For example, you'll often hear this incorrect use of 'd better': " I like winter sports. I enjoy skiing but I'd better go snowboarding. That's my favourite sport". This speaker has clearly misunderstood the meaning of 'had better', or else been incorrectly taught'. The correct way to say this would be 'I'd rather go snowboarding'.
The only meaning of "I'd better (do this)" is 'I really should do this'. This means that there will be negative consequences if I don't do it. For example, "I'd better go now" means 'I need to go now' (because if I don't go now, I'll miss my bus and I'll be late for work, for example). It can never mean 'prefer'.
4. Don't believe any native speakers who tell you that it's correct to say "You better" without the 'd'. This is an informal, non-standard form which is not generally used by educated native speakers. You will hear this, but it is not considered to be correct in standard English.