It seems to me you are asking two different questions here. One is in the title: "How to clearly communicate that you're asking someone on a date?"
Well, that one is easy. Just tell them "Hey, I'm romantically interested in you, want to date?"
But, in fact, this isn't the social appropriate way of doing it, because, as stated in the comments, there's a lot going on about not losing your face or make your counterpart lose theirs, and therefore everything has to be veiled until the relationship is stable enough to ask/start even very intimate questions/actions. One way to feel your ground during this span of time is, again, as mentioned, to start small physical contact like touching hands, forearm, shoulders, holding hands, looking each other deep into the eyes...
But all this would be the answer to "How to convey to the other person you see this as a date". So, in fact, I'd say there is no socially acceptable way to communicate a date as a "date" that couldn't turn into something very awkward. Like, reaching for the other partners hand, holding it between yours, looking deep into the others eyes and whispering in a very promising voice "Want to go on a date with me?"
Well, on to your second question: "Was there anything that I could have done from the start to avoid this situation?"
That's a hard one, with me not having been there! As Tinkeringbelle already explained, this could have been a case of mistaken intentions on the other's part. It could also be a case of not clearly transported intentions of your part. But, and that's the third, it could also be a case of change of mind. Or, the fourth, a last test.
Let's first have a look at the "last test". So, there was this great date, you feel just very confident, and along comes this icebucket of a full stop. What are you doing now? Will you go into sulking, will you throw a temper tantrum, will you show her the bill for the evening and demand instant splitting? You'd be surprised how many people see an invitation for a date as an investion which gives them rights.
Or, will you stay decent and honest and answer something on the lines of "Well, for me it certainly was. You sure it wasn't?" Which would open the ground for the other to speak their mind (or make it up). After which you really should know where you stand.
For the change of mind, well... there a a billion things that could just spontaneously tip the scale against you, even on a date going well up to this point. Maybe it was a date, but then you tipped your waiter too much/too few/too whatever for their like? I don't know. You could, if you were reading their body language correctly. Or not, if they are very good at acting. See, so many ifs and maybes...
It could even be your date is one of these people who like to be courted without commiting and therefore wants to prolong this stage of uncertainty and wooing indefinitely.
So, in fact, the second question also can't be answered with certainty. Because there are two of you in it, and the other one surely has an important part to play.
So as I see it, you have two general options. The first is to take the ending of your date as a "no" and act however you act when rejected.
The second would be to take this as an invitation to turn the situation into something definitely a romantic date. Since it is too late for you to do so with mentioned date, ask them out for a second, maybe with a wording of "Would you like to go on a definite FIRST date with me?" When they agree, see it as an open invitation to further the physical contact, touch them on all these little opportunities, and certainly try to kiss them goodbye. And watch their reaction to each of these interactions. Whenever they shy back from them, you plan hasn't worked...