How to finish a polyurethane coat

When you apply multiple coats of urethane, any dust or small flaws will be amplified with each new coat. If you want the smoothest finish possible, then wait until it is dry, very lightly sand with very fine paper (220 or finer) to remove any dust bumps, wipe it clean with a slightly damp cloth or tact rag, let it dry or buff it dry with a dry cloth, then apply the next coat. There is no reason to use mineral spirits, actually I would advise you not to use mineral spirits.

I see no advantage to applying successive coats with, then against the grain. As long as you apply an even coat with a good quality brush or foam brush and don't overwork it, any brush marks should disappear. The finished product should be so smooth that you couldn't tell which way you brushed the urethane on. Also, if you are using a satin finish urethane, be sure to mix it well, stirred, not shaken. The dulling agents tend to fall to the bottom, so it is important to mix it well. No matter how many coats you apply, it should not become "glossy".

You shouldn't need to strip unless you scratched through to wood.

The problem with polyurethanes is that they are hard films that don't blend with the previous coat. Other finishes will 'melt into" the previous one (lacquer, shellac, tung oil)

Your sanding, @220, was too aggressive. Intercoat sands should be at 320 - 400.

You will have to sand back to a common level that eliminates the scratches and lines. If you leave part of one coat and part of another coat, there will be a "witness line" . You can wipe down the area after sanding with a soft-mineral spirits cloth and look for the lines.

Be sure you are allowing enough time between coats with the poly.