or does my LCD monitor have to be HD to like the full visual display of HD?


Answer:
True HD resolution for 720p is 1280×720, so in this case technically, no it's not 'true' HD.

Keep in mind however, that Monitors (for a long time now) have been able to produce superior resolution than most Televisions (1280×1024 is a common resolution for most LCD panels which is better than 1280×720 obviously (the extra height is because the resolution is designed for a 4:3 (square) monitor)).