It is important to understand the requirements of each of the arguments of the different functions you are using.
In the IF function, the first argument requires a complete mathematical expression that results in a True/False value to be returned.
That can be accomplished by writing an equality/inequality formula, or by using a function that returns a boolean function (such as IsOdd, IsError, etc).
So:
A1<="09:00"
A1<=(9/24)
A1<=TIMEVALUE("09:00")
are all valid boolean mathematical formulas (though they do not all do the same thing, as you discovered).
A1"<=09:00"
is NOT a valid formula. Remember, anything enclosed in quotes is treated as literal text. So in this instance, the "<=" is treated as text, not as an operator.
However, the COUNTIF function operates differently. The second argument does NOT want a complete formula that returns a boolean value.
All it wants is the condition of the formula (to be applied to the range in the first argument). It is not looking for the whole formula, only the condition, and it wants that condition returns as text.
So:
">=90%"
"<=99.99%"
are valid conditions, but not complete formulas.
Since you can build the conditions dynamically, the literal parts are in double-quotes.
For example, instead of seeing if the values in your designated range are >=90%, suppose that you wanted to see if they were greater than or equal than the number in cell D1. Then you could write the condition like this:
">=" & D1
Keep in mind that Excel has pretty good documentation on all their functions (the formula helper is useful), and there is lots of great information out there on the Web too, where you can find details explanations and examples of all these functions.
One final note on your original function:
=IF(A1<=(9/24),"TRUE","FALSE")
Since you have "TRUE" and "FALSE" in double-quotes, this will return the text string TRUE and FALSE. If you wanted to return the boolean values, you would drop the quotes around the TRUE/FALSE. And actually, if you wanted to do that, you don't even need the IF statement. You could write it as a straight-up boolen function like this:
=A1<=(9/24)
If that statement is True, it will return TRUE. If it is False, it will return FALSE.
Hope that helps.