Why are you mixing tempo with T.S.'s? They have nothing to do with each other. T.S. is the feel and tempo indicates the pace from beat to beat. The pace of the song NEVER effects the T.S. Never.

The difference is exactly what you said. You only hear a clear pulse on 1 as you ascribed. Then 2 beats follow, although there is no clear pulse. Your 1 2+3+ is more like counting "One ta-ta-ta-ta" rather than "One Two and Three and"? Right? Two and three don't really have a pulse. So it can't be 3/4. There is no stress on two and three, thus two and three (as you wrote them) are actually a background of One. A compound time signature. Maybe a better argument would be to say it's 3/8 instead of 3/4.

Alot of people don't understand time signatures. Especially Compound time signatures because the top number (6 of 6/8) doesn't indictate that there are 6 beats. Instead the division of a beat, 6/8 actually has 2 beats. Each divisable by 3. Which adds up to 6. Kinda complex but it does help in the long run of understanding feel.