I use only internal cache crawler because external crawlers (like Netpeak Spider) not generate cache or lifetime of generated cache is very small (not sure)
but if I previously generate cache by internal crawler then external crawler says "quick response".
Really? If so, something must be wrong with defined cache parameters or/and cache varies. Anyway, I am out from here and can't help anymore. To solve your issue someone must have access to your installation.
No, I don't think so. In my mind the most important and the very first way would be to isolate the reason and to find out, if the issue only happens with crawler or independently from it. For this it would be very very very important to use same UA for crawler and for manual request with browser. Otherwise you will always fail.
Independently from this I remember to many of your posts in the last 2 years. You did a lot of custom modifications almost everywhere. Maybe the reason for the issue is sitting in front of the monitor....?!
Finally and to extend your knowledge about economically cache warmup, read the following documentations. There is definitly no need to warmup the cache for Google bot and/or to define cache rules for Google or any other bot to get better Pagespeed score. It may irritate, but I have disabled cache function for any bot!
You can instruct the Cache Engine with cache-control environment variables in .htaccess. Just be awarethat X-LiteSpeed-Cache-Control response headerswill overrule any settings defined by rewrite rules.
First of all you have to know is, that Pagespeed doesn't measure the loading time. It measures the "display time" after all data has been loaded from the server. LiteSpeed cache is speeding up the loading time, but not the display time, because only the php based main document is affected from caching but not static sources like .css or .js. These static sources and the way how the are loaded affect the display time Pagespeed is interested in. For Pagespeed it doesn't matter if a HTTP cache is used or not, because and again Pagespeed can't and doesn't measures the loading time.
If you allegedly have disabled caching for bots and if you get bad results now some other things must be wrong with your cache varies. Litespeed cache has almost no effect to Pagespeed.
Sorry, I can't answer your last post. I don't know what cache varies you have defined, but if you think something is wrong and it doesn't work as you believe it should, check your cache varies. There is definitely something wrong with it!!