Yes, you can disagree all you want, and defend your design with the level of resolve a professional should have. If one person’s opinion were all it took to send you careening off in some other direction, your days as a designer would devolve into a chaotic spiral to failure.
There are a thousand companies with 3-stripe logos that Adidas hasn’t sued. And arguably, your design doesn’t even have three stripes.
I see very little resemblance between the Adidas and Link logos.
However, I don’t think the relevant question is whether or not the logos look too much alike (they don’t). The question, in my opinion, is whether or not Adidas might sue. They’re both sports/fitness brands, European, and Adidas has a well-deserved reputation for suing other companies. If you look at the Thom Browne link Smurf2 provided, you’ll see no particular resemblance between the Adidas logo and what Thom Browne has, yet Adidas still sued them. I doubt Adidas will prevail in their lawsuit unless there’s more to it than a stretched argument over the use of superficially similar stripes, but it’s still a huge hassle for Thom Browne.
So, yeah, maybe Smurf2 brought up a legitimate concern. In my opinion, it’s ridiculous that such a concern exists, but given Adidas’s reputation, it’s still something to consider.
This sort of concern, by the way, is one of the half dozen or so reasons I hate designing logos.
As for the word mark…
I already mentioned my trivial reservations, so let me explain my equally trivial solutions. If you enlarge the word mark, you’ll notice the top and bottom corners of the diagonal strokes on the N and the K are slightly curved. If it were me, I’d exaggerate these curves a bit more to make them better match the curves in the diagonal strokes of the logo.
In addition, I’d reduce the size of the giant gap between the L and the I. I’d do this by shortening the horizontal stroke on the L, then moving the letter slightly closer to the I.
The first logo grabbed my attentions right away, I think it looks strong and I don’t see any connection between your logo and Adidas at all.
Firstly from an aesthetic perspective I like the bold, clean mark, well done.
Usually when I’m struggling to determine whether a piece of design is appropriate, it’s often the result of not having enough clarity of the intended target. In your case I think your target audience is too broad and that you need to narrow it down a bit.
You say it’s for people that are goal-oriented, what are their goals? The are so many reasons people hit the gym: are they powerlifters 
, are they professional athetles or just regular gym goers trying to stay in shape?
And who exactly are these people: are they affluent or poor? What kind of car do they drive? What are their interests? Do they play any sports?
There’s probably a very good reason for this: red is quite a confident, passionate colour. Why do you think this brand needs to look different?
You say it’s for people that are goal-oriented, what are their goals? The are so many reasons people hit the gym: are they powerlifters
And who exactly are these people: are they affluent or poor? What kind of car do they drive? What are their interests? Do they play any sports?
That’s the hardest part of this project, online shop targeted for home workout, but it can be both heavy and light equipment. I need to hit both. Logically my target is people of avarage and above avarage income. I’m not sure how to define target audience more in detail.
There’s probably a very good reason for this: red is quite a confident, passionate colour. Why do you think this brand needs to look different?
Just to stand out in the competition. I think there are more colors suitable for this type of brand, not only red.
Maybe ask your client these questions - What kind of market research have they done to validate there’s a demand for home exercise equipment and who do they anticipate will buy it?
Price-wise where do they sit in relation to their competitors: are they selling affordable equipment or is it high-end and expensive?
These are very precise and well-defined questions. It’s like something I definitely want to know, but couldn’t define due to lack of experience (it’s my first client). Thanks, it will help on this project and in general, for future researches.
Why don’t you try the link as in the chain link? Would be stronger than a fitness rope ;-D
Explain how a chain represents fitness, when ropes obviously do.
Made me think of screws. Might just be me though.
It’s been 5 months since the OP posted this. I’m pretty sure this project is done.