In Japanese, is it acceptable to write kanji letters in the Chinese style? For example, is it acceptable to write 田 with the 3rd and 4th strokes swapped, or 着 with the 6th and 7th strokes combined?
I feel that at the extremes of stroke order perfection foreigners always seem to be better than native Japanese, maybe because there are just too many who make it a pastime to know all stroke orders for all sorts of obscure 漢字. Unless you are dealing with a 書道 teacher (or school teacher), the general focus is more on whether you are able to remember all strokes correctly, no matter what order. Now 田 is a primary two 漢字 and an exception to one of the first rules "if a horizontal line crosses a vertical line, the vertical line is written first, by which it would be more sensible to write 冂 > 土 and not 冂 > ｜ > 二. But because it is the archetypal exception to stroke order, I would say that all Japanese do remember 田, but a few might get caught up in 荘 or 我, especially if you make them think about it. (The Japanese obviously will have a better kinetic memory than you. You might enjoy this question about 空書.)
Stroke order is one thing, but combining strokes is practical and the foundation for most styles of 書道. Combining strokes is what must happen naturally. (They are still conceptually 12 strokes in a particular order, even if you didn't lift your pen/brush between stroke 6 and 7.)
In any case you are still wondering, regarding the reaction especially to the first blunder, I think @ssb put it quite well in his comment.
Yes, it is not that big a deal to get some ordering wrong. Most of the time, unless someone is watcing you write, they won't even be able to tell how you wrote it!
The reason the stroke order is emphasized in Japanese schools is that as you start writing kanjis faster and faster, strokes start to join together. You can see the most beautiful example of this in 行書, but you see some of this in normal hand-writing people do. When you are writing such text, getting the stroke order right becomes critical, as others won't be able to read it if the order is diffrent. Similarly, to read such letters one needs to know the proper order of strokes.
As a foreigner studying Japanese, you should be sticking to 楷書, with each stroke cleanly separated. So long as you do that, don't worry too much about the stroke order.
Most of the Japanese text that is produced is produced by printing (books, newspapers, magazines, office printing, electronic displays, ...) and in printing there is no stroke order, or no evidence of stroke order in the end-result.
In my Japanese language school, there were a few Chinese students, and my teachers would emphasize before exams that Kanji have to be written using the Japanese stroke order.
On the other hand, for the character 備 they insisted that stroke 7 starts where stroke 4 ends and not where stroke 6 starts, even though in my understanding, the former is the ("modern" traditional) Chinese stroke order, whereas the latter is the Japanese stroke order (as well as the "traditional" traditional Chinese stroke order). Compare 備 in the wiktionary and 備 in the Kangxi (p. 113 character 21).