All name servers eventually will have the same records right? For example, initially only Namecheap DNS servers know about my website. Eventually, DNS servers all over the world will mirror the contents of the Namecheap server and vice versa?
I think walking you through the steps will help you understand a bit better.
1. I type the name of your website - example.com - into my browser. My browser initiates a lookup with my Operating System, which contacts my preferred DNS server. I can either program the IP address of this DNS server into my OS, or program it into my router, or let my router pick it up automatically from my ISP when the router/gateway gets its IP address from the ISP upon connecting. Usually there will be a caching DNS server maintained by my ISP not too far away, in terms of network jumps, from my connection. Let's assume this is the caching server used in this case.
2. If the caching server has recently looked up the DNS entry for example.com, it already has it cached. It sends it back down the line to your browser. Done.
3. If the caching server does not have this DNS entry in its cache, it will look to the TLD server for the TLD in question. TLD = Top Level Domain. .com is a TLD, .net is a TLD, .us is a TLD, etc. These TLD servers keep track of EVERY domain under the TLD, but they don't keep track of the actual IP address for example.com; they only know which nameservers are considered authoritative for example.com. In this case, you are using Namecheap's DNS servers as your authoritative DNS servers for example.com. The TLD servers send the IP addresses for Namecheap's DNS servers back to your ISP's caching server, which then sends a request to those Namecheap DNS servers for the location of example.com, which should be answered promptly and forwarded back to your PC (and cached by your ISP's caching server for 24 hours, 48 hours or whatever time limit is specified by the Namecheap servers).
4. Finally, having obtained the IP address for example.com, your browser initiates the HTTP request for the website at example.com.
So... no, all the nameservers in the world will not ever have a copy of your info (unless your website is extremely successful like google, yahoo, etc.). Whenever a user visits your website, their local caching DNS server (and any intermediate caching servers up through the ISP's hierarchy) will cache the results for the specified time to live (TTL). Once the TTL is up, the result is deleted and will have to be queried again from the TLD server, which points to your nameserver and not the site itself.
Of course, that is just the "official" version of how DNS works. I would bet that most big ISP's basically try to keep a live cache of pretty much all the big TLD servers, at least the ones relevant to their country of origin. I also think it's likely that most big ISP's have several layers of DNS cache hierarchy (lots of servers close to end-users, which deal directly with fewer servers higher up that maintain bigger caches), so no one server gets hit too hard at one time.