Hi. If users do not use the AMP version they will not use it almost automatically. What is the use of an AMP version if it is not used automatically?
The use of AMP is essentially search engines related. AMP results will have better visibility and better conversion than common responsive websites. Then on some sites, if the feature set is ok, it can be used to totally replace both desktop and mobile site (what we call "standalone" mode). And for those who have very bad mobile site, yes, it can replace the mobile site.
automatically my phone started the AMP version of the site
This is bad and should not happen automatically. In addition, I do not know any plugin which does that automatically. Actually, the most common WordPress AMP plugin, like us, does not allow showing AMP version to phones.
All this is because my template in normal version is very good and I have it very optimized. I have a score of 95 on Google Page Speed. The mobile speed test is very bad 36.
I would like to combine the two things, when a computer in between shows the good template and when a mobile AMP version.
The Google Page speed is only a raw indicator. Is your site slow? That's what you should measure. Actual speed numbers. Your site may be considered a little slow indeed on 3G network, mostly because you use large images. Have they been fully optimized?
Is it possible by modifying something that the mobile phones start the AMP version automatically?
That would be a rather large change, not something you can do with a few lines of code. We have that on our list and we'll do it at some point but as it's not in high demand, we cannot provide an ETA on it unfortunately.