In Arabic short and long vowel refers to the length of time, not necessarily a change in the sound of the letter. Here's where it gets tricky. In native Arabic words, vowels are only written out if there is a long vowel. For example, the neighborhood I used to live in Westcott, if it were a native Arabic word, would be written wstcot (double consonants are also not written out). Foreign words are spelled out with all the vowels as if they were long, since one couldn't just look at the word and recognize it. Short vowels are written in certain cases: if it's a legal document or a religious text where they want zero ambiguity about the meaning, or a new vocabulary word being introduced to students. In these cases the symbols on the top left of that chart in the last post are used: the Fatta (short A sound), Damma (short U sound), and Kasra (short I sound). In the photo they are shown in relation to the letter Baa.
Lastly, the Taa Marbutta! This is a letter that comes at the end of a word, and makes an "ah" sound in Egyptian Arabic, and (usually) and "ih" sound in Levantine. You see it in words like حفلة (dance party) and my name فرحة (joy/Joy). It indicates a feminine noun, and this is why you will hear Egyptians invite you to a "hafla" while Lebanese and Palestinians will invite you to the "hafli".
Lastly, I should mention that the long vowels II and UU, if they are next to another vowel OR if they are at the start of a word, change their sounds to Y and W, respectively. So a word like شوية "Shwayya" (a little bit) might be spelled in English "shuwaya" or "shwayya" "shwaeea" etc. When you hear a word that starts with one of these sounds, like "olooloo" (tell him) or "Ibrahim" the sound is a short vowel riding on an Alif at the start of the word. This is a spelling thing, and doesn't usually affect transliteration too much, since the sound its self is pretty clear.
Next post we'll talk about consonants, and those sounds that the Arabic language uses but we don't have equivalents for in English. You'll find out why some words are spelled with numbers!