Depending on the dialect, British English is often very slang infused compared to more international varieties (American, Canadian, Australian, etc.). The situation with British dialects is perhaps a bit like Germany, where local dialects have developed over several centuries longer than in Britain’s former colonies. I think some dialects may shorten the ‘a’ sound a little bit so it might be confusing, but I have never heard of “back” coming out like “buck.”
As far as I know, typical pronunciation would always or almost always be how @livelaughlove21 suggests. That would be similar to the first letter in German Apfel or the ‘a’ in Straße. The Dutch short ‘a’ like in maan (Eng.: “moon,” German “Mond”) is not so far off either. I doubt anyone would say you are wrong if you took the ‘a’ from either Apfel or Straße and put it in most common words that have a short ‘a’ in them: back, hat, band, hand, cat, ran. You are not doing anything wrong if you articulate the ‘a’ more strongly than native speakers. I think the ‘a’ in Straße as this guy pronounces it be closer. To my ear, the ‘a’ in Apfel usually sounds a bit more articulated, but both these vowels are very similar to an English short ‘a’.
Also to correct one of your sentences: I noticed that we learned English at our school very differently than the people speak it. It is a good sentence to practice that ‘a’ sound in different words. American students often confuse then and than.
Or a shorter sentence: A cat sat at the vat. Four of those words should rhyme, and in my dialect of English the the indefinite article is a short ‘a’ as well.