Starting from scratch:
- "Manure" refers to organic material used as fertilizer. This includes animal manure, compost, and green manure.
- "Compost" is decomposed organic matter.
- "Tea" in this context is the liquid resulting from steeping compost (or manure) in water for a period of time.
- "Leachate" is liquid that leaches out of something.
Given the definitions above, compost is manure, and broken-down animal manure is compost. Most people I know who say "manure" mean animal manure -- any other use is qualified, e.g. "green manure". "Organic" regulations in some places may make a technical distinction between "manure" and "compost".
Below when I say "manure" I mean "animal manure", and "compost" means compost produced without any animal feces.
- "Compost tea" is the liquid product created by steeping compost in water.
- "Manure tea" is the liquid product created by steeping manure in water.
- "Compost leachate" is the liquid that seeps from the bottom of your pile or tumbler.
- "Herbal tea" is something I'd make in the kitchen to drink... I'm not sure about this usage in a gardening context except to assume that it involves steeping herbs in water to use as a spray for plants...
The leachate may have high concentrations of pathogens if they are present in your pile. I wouldn't spray it on my vegetable garden. I'd just let it leach into the soil around the pile and enrich the soil there -- it is likely to contain nutrients that leach out of the compost pile. Covering your pile can reduce the amount of leachate produced and thus the amount of nutrients lost from the end product.
I'd only make manure tea with well-rotted (composted) manure to avoid contamination with pathogens -- or use it only on non-edibles.
Compost tea is useful as a fertilizer. Research indicates it may help fight certain plant diseases.
Finally, just like compost has many "recipes" and techniques, and widely varying quality levels, there are many recipes and techniques for creating teas, each with widely varying quality levels.