Is this an important and necessary distinction, or would it also be
correct to write 有 with the thinning out, bent first stroke, like in
月? And would it be correct to write 月 without bend and thinning out?
A short answer to this question is that these distinctions will never result in something considered a different character, and at worst, could just be considered a quirk of individual handwriting. The full answer is more complex.
The writing of a character can differ according to script (e.g., Regular Script 楷书 vs. Clerical Script 隶书), orthodox forms vs. variants (e.g., 靑 vs. 青), and font.
Here is an image from Wikipedia showing several variations ( From left to right: sans-serif (gothic), Ming, regular script, clerical script and seal script):
Note that regular script characters often do not differ in structure from the equivalent Clerical script characters, but the "rules" for forming the strokes and the overall proportions of the characters differ.
Variations in characters you will encounter will vary between personal calligraphical variants, structural variants, font variants, and variants that combine these characteristics. To decide which variant is right for your handwriting, you have to pick which authority to follow. There are three categories of choices.
From a calligraphic point of view, people look to famous calligraphers for examples. These calligraphers may differ from each other, but often present somewhat of a consensus at a certain level of detail. At this level, you have to consider more than just whether a stroke is Piě 撇 or a Shù, 竖, but the precise direction and curvature of the stroke. The Shù 竖 in 十 is usually precisely vertical; whereas the Shù 竖 that starts 口 can be angled inward slightly in good handwriting. Similarly, the Piě 撇 in 月, in 大, at the top of 奈, and at the left of 龙 all vary slightly.
In 有, the second stroke is definitely Piě 撇; but the third stroke, whatever it is called, can be written with a slight curve to the left that is less pronounced than the curve in 月 by itself. On the other hand, in the Ming font used at the top of the Wiktionary entry for 有, the third stroke is perfectly vertical and indistinguishable from the first stroke in 口. At this level of detail, notice that the last two horizontal strokes in 有 do not go all the way across in the handwriting example further down the page:
Also note that the last stroke in 口 kǒu and 囗 wéi can be written differently, with the first joining the bottom line from the top and the second joining it from the bottom with a hook:
The character forms taught to school children are prescribed by the regional and national authorities in Mainland China, Taiwan, Hong Kong, Japan, and Korea; however, these authorities often do not agree on the same variants, even if we limit the selection to the "Traditional Script." Some appeal to the Kangxi Dictionary as the most authoritative source, but its "orthodox" versions contain enough of its own quirks that nobody completely follows it.
Adults generally write in a cursive that includes features and substitutions common to everyone, but with details that vary from person to person. In writing formal Regular Script with pen or pencil, I think people try to follow the normal brush shapes, but ignore issues of thickness and thinness and other details not easily reproduceable without a brush. I once saw a book that taught calligraphic stroke with a ballpoint pen, but I think it included many details most people would not bother to respect in normal writing.
For a beginner, the best thing to do is to find a reliable authority for stroke order and stroke type, preferably showing handwritten versions. If all you have are printed characters, just live with the fact that your characters may contain the odd quirk here and there, but still be easily recognizable. Don't sweat details that do not clearly differentiate different characters. No matter what you do, there will probably always be a more calligraphically correct version than what you learn and reproduce, but it is not worth bothering with all the details when you are just beginning.