Knife or pen grip when crocheting? Is one more defined?

Hi, it’s me again, this time with a slightly different question.
Ever since I started crocheting, I’ve used the knife grip (holding the hook like a knife). Since I’m currently translating a lot of patterns, I noticed through a friend that her crochet work looks extremely defined—in other words, the individual stitches crocheted (in this case in X-stitch) are extremely clear and distinct. I’ll attach a picture.
Now, about my experience: first of all, I’m currently using Schachenmayr Catania, and I was previously crocheting with a 3.5 mm hook. To make it more defined, I’ve now gone down to 2.5 mm. Despite that, it still doesn’t look the way I want it to, so I took a closer look and this person crochets using the pen grip. I actually thought it wouldn’t make a difference, but when I tried it myself using that grip, it somehow looked more defined.
The problem now, though, is that it feels like I have to learn to crochet all over again. I’m extremely slow and it takes me an extremely long time. Am I right in assuming that this might have something to do with the grip, or could there be other reasons?
I hope you can help me here...
Best regards
Moe

P.S.: The picture below shows how I’d like it to look...it looks extremely even and tight.

Mod Edit: Image removed at the user’s request.

Hi Moe,
is this your picture? Please confirm briefly, otherwise I’ll have to remove it.

This isn’t my picture, but I have permission to use it

OK, thanks for the reply.
Could you please note in each post whether the image belongs to you or whether you have the rights to it? Having to ask this over and over again really takes a lot of time. 

I actually can’t imagine that the way you hold the hook would make a difference. I’d guess it has more to do with the hook’s thickness.
I’ve already tried both, but I’m more comfortable with the “knife grip.”

Could it be that with the “pen grip,” you’re pulling the working yarn from a different direction? For example, with single crochet, do you pull the working yarn through the loops from below with the “knife grip” and from above with the “pen grip”? This changes the stitch pattern significantly.