Hi @skatofiabah ,
First, sorry for the late reply.
As far as I can tell, 99% of my paid search comes directly from Google (google.com and google.ca)... I have my paid search detection set up to look for search engines with gclid or utm_medium matching one of our know values (paid, cpc, ppc). It shouldn't matter if you use CID or UTM really (almost 95% of our paid search is missing our UTMs... despite me telling that team multiple times to fix it...)
I have 0.2% coming from https://syndicatedsearch.goog (this being my top 3....)... 0.2% is so small, it barely registers. It looks like I don't have traffic coming from that domain for other uses... so I guess that one is probably safe to say "Paid Search", but it's really not going to catch the bulk of your paid search traffic.... As for https://googleads.g.doubleclick.net/ I would say this is 100% ads (big box, skyscraper, leaderboard, etc) and nothing related to paid search.
Both Organic and Paid Search are generally going to share the same referring domain... this is why Adobe has the Paid Search Detection settings per suite:

Adobe knows what referrers should be search engines, then you can specify what query parameters are being used on your paid search.
The referrals with the specified parameters will be grouped as "paid search", and everything else as "natural search"
If you are trying to use Crazy Egg (without access to your Adobe data), that might be a bit harder. I have never used it... and if you can only filter by referrer, you won't have enough detail to split that into Paid or Natural Search... (since it's the parameters in the URL of your first page that determines if it was paid or not....)