You do have lounge access when travelling internationally. Do you have an international segment AFTER your flight to AMS? Note that DL, in contrast to common belief, is not understanding a flight from TXL, Germany to AMS, Netherlands as international travel but comprehends international travel as: "Customers traveling to/from Europe, Asia, South America, Central America, Africa, Canada, and Mexico (excludes travel to/from the Caribbean).".
Sometimes they won't let you in even if you have a TATL portion because they do not recognize your GM card, just point to their little hint book that does indeed shows the SM Gold card now.
However, the lounge is quite basic and KLM is usually flying from the little terminal beside the main gates, so you need to leave early enough to make it through security there.
Oh, please. DL's website is dumb in that regard, but SkyTeam rules prevail here and a GM will have access on TXL-AMS. I have no issues as a PM (nor did I as a GM) using ST lounges in Europe when flying intra-European cross border flights. Not even a problem if flights are within the Schengen zone.