By the nature of the question, we are pretty much narrowing it down to modern presidents, since few people actually got to see the presidents give speeches prior to the invention. There are 2 that come to mind and are fairly recent: Reagan and Clinton.
You have to love a guy that gets elected and the Russians increase their readiness for war level because of his inauguration. That is highly influential! Maybe not in a good way, but influential to say the least. One of the largest landslide victories in US histories started his first term as president, as many life-long democrats voted for him instead of Carter. I can say this as it was the only time my parents actually voted for a republican and they talked about how hard of a decision it was for them. He set the standard for how to deliver bad/important news to the country. It made you actually feel as though you were in the process...that he was really reporting to you...the people that elected him. How many times have you heard it replayed: Mr. Gorbachev, tear down this wall! He made you feel good and unashamed to be an American.
Clinton was a extraordinarily media savvy president. No one had ever heard of the governor of Arkansas, but then the media's spotlite hit him and never lost him for 8 years (still continues to make news to this very day). Think of the last time you heard a meaningful Carter, Bush Sr. or Ford comment. Not sure if it is still true, but after his final term, Clinton had dropped more bombs than any other president in US history. Interestingly enough, he is known as a peace-maker. He always looked comfortable in front of the camera (save for a few Monica interviews) and seemed as though he was one of "us" asked to do a very difficult job and that he was doing it the best he could. He ate CHEESEBURGERS...just like you and me. I always have a picture of him running in that jogging outfit down to the local DC Mc Donalds. If you ask people in other countries (and I have), "who was the greatest US president?" and you will most likely hear his name mentioned at the top.
So unfortunately, I have to give the nod to 2 of them. I think for eloquence as a US citizen, it would be Reagan. However, if you are viewing from outside, then the answer would be Clinton.