Time signatures are primarily for notational purposes. Beat, tempo, and meter all describe a certain thing about the music, but the time signature is just how that's codified when it's written down.
As you know, Tempo is the frequency of the beat, and Beats are a kind of rhythmic emphasis that happens at regular intervals in most music. Meter is an important term--that tells us how each beat is subdivided (i.e. in two or in three) and how to group beats into larger chunks, also at regular intervals.
Time signatures, plus information about tempo, tells us how to get all of that information from notes written on the page. A quarter note alone has very little meaning until we contextualize it in terms of what note value the beat is assigned to and what the frequency of beats is.
You are correct in noticing that there is overlap between tempo markings and time signatures. That is because, while there are some conventions for how they usually relate, these conventions can be broken.
Consider 4/4 time with a tempo marking of q = 60 (bpm). This one is simple, there are sixty quarter notes per minute, and four quarter notes per measure. But what if the tempo marking was h (half note) = 60? There's no hard and fast rule that says the tempo marking has to be equivalent to the time signature (and in fact in 6/8, it rarely is), but in that case, there are still four quarter notes to each measure, it's just that the tempo is being given in how many half notes there are per minute.
Tempo is usually given with a note duration. If that duration is missing, then you can usually assume that it is referring to the bottom part of the time signature. But it is fairly rare to just see a number at the top of the page without any context. In the example you give, there is no note value assigned to the beat, so you would assume that it is referring to the quarter note for time signatures of 4/4 and 2/4.
For 6/8, things get a little bit hairy. This time signature always means that there are six eighth notes per measure, but not always does the eighth note get the beat! 6/8 is most commonly used to refer to a duple meter with a triple subdivision. This means that a dotted quarter note would get the beat, and each beat would be divided into three eighth notes. Most American folk songs fit this model (think about how you would write down "Pop Goes the Weasel"), and 6/8 time is used to avoid writing a triplet figure over every subdivided beat. 6/8 tempo markings are usually written as q. (dotted quarter) = x, but they occasionally will be e (eighth note) = x, signifying that you have six beats per measure.
