You are indeed right, these two signs (+) and (-) have confused generations of scientists. I have searched for more than 10 years on Volta's assignment on the signs of battery terminals. I will summarize my findings below. What is disappointing when these chemistry or physics teachers say...well, it was just a convention. Sorry, this is not true, scientists from the previous centuries were as good as any other scientists today. There is a very valid reason why they labelled the batteries that way. What confuses most people is the transition from the signs of pure electrostatics to current flow-like a battery. How did the batteries get their labels?
Forget about electrons for the time being. This is a relatively recent discovery. Let say you are in the 18th century. You will have to follow the train of the thought of these early physicists. First of all, you might already be aware that Benjamin Franklin called one type of charge as positive and the other type of charge as negative. You have already got the details by Conifold by the time I was typing the answer. The functional definition is positive charge is that it is the charge that is obtained by rubbing glass with silk. Label the glass rod as positive electricity. Rub the resinous bodies, and you get negative electricity. This is the functional definition of the negative electricity.
The key point is that all electrical signs on batteries can now be traced these functional definitions given above. Keep in mind we still do not know what an electron is. Now there is a device called the condensing electroscope. Volta himself invented it. This can sense very small charges, as found in batteries, and causes the gold leaves to diverge. Let us say you have an unknown charged body, you would connect the object to be tested with the condensing electroscope and charge the electroscope. The leaves will diverge. Note that ordinary electroscope won't show any response. Next bring your charged glass rod in contact with the condensing electroscope. If the leaves diverge further, your test body had the same positive electricity. Your test body will get a (+) sign. In same way you can test it with a rubbed resinous body and if the leaves diverge your test body will get a (-) sign.
So far there is no need to invoke electrons and protons. Now you may guess that how Volta would have tested the sign of the ends of this pile using his condensing electroscopes. Note that electrometers were a fancier version of determining the signs of charged objects. This term will occur in the suggested paper below.
With that you should this 1930's paper from Nature (Open access) in order to find the answer to your query The Original Mode of Constructing a Voltaic Pile. It is a single page article but it will clarify the junk our modern textbooks have created about the signs of the Voltaic pile.
Food for thought:
a) Rarely any physics or chemistry textbook talks about about determining the sign of the electron's charge? How was this determined I will let you find out. Hint, it is also traceable to resinous electricity definition.
b) How did the early scientists find out that the charge carriers in wires are negatively charged particles?