Yes that site is safe, but so are all of the sites that are TD's (Trusted Dealers) They are on that list for a reason..... They are Trusted.
I know it is a little different when you are new, but trust some of us that have been here for years when we say that there is a reason a dealer is on or not on that list. There is literally no way a dealer gets on that list without lots of vetting etc. It does happen occasionally that one of the TD's gets taken off the list, but that is rare.
Now that we have established that any of the TD's on the list is just fine to buy from, the question becomes which ONE? This is where you have to put in a little effort. Every TD buys from the same source and basically none of them carry inventory so they can all get the same watches. There is a difference though. They do have different policies on after the sale customer support. A few will not help you out if the watch gets seized by Customs and others will do a 1 time replacement. Others might do unlimited replacement etc. Some will offer to fix a "dead on arrival" watch & others will not etc. As you can imagine, prices will vary somewhat by what services they offer. Another big difference is in how they communicate and in how closely they inspect a specific watch for you. The bigger shops like TrustyTime, PureTime, InTime, Toro Bravo etc have beautiful websites with really nice pictures, but they are bigger shops, more automated and in my (and many other members) opinion do not offer the best communication nor do they do the best job of being really picky when picking out a watch for you. Other smaller shops like Lucy @ TrueTime168 and Sead @ Supermirrors and Mary will do a better job of communicating and do a better job of QC. You might have to wait a bit longer too, but in my opinion it is worth it.
Have fun and good luck with your hunt for the perfect watch.