I believe JMac is correct that Adam's scars are a continuation of the electrical burn on the Stranger's (i.e. middle aged Jonas') arm, but I wanted to add an additional consideration.
There are three ways to travel:
- Using a machine that stays behind (Adam's device, Noah's chair, the 2053 power plant)
- Using a device that travels with you (the suitcase/golden ball)
- Using the "natural" tunnels in the Winden caves
It's important to note that the suitcase/ball travel with the traveler, but the static devices such as Adam's machine and Noah's chair stay behind. Therefore, someone travelling using Adam's device or Noah chair cannot make it back easily, since they have no time machine anymore.
Jonas, having landed in 1888, has no way to travel anymore - at least not without being unable to make it back. He eventually created a time machine, but it doesn't travel with the traveler, so if he were to use it himself, he couldn't get back without using the suitcase/chair/golden ball, which he doesn't have.
So it seems Jonas/Adam never travels away from the 1888-to-1921 period, except at the end when he never returns (edit I omitted the reasons for his travel as I'm not sure if you've seen the series finale). In any case, Adam only travels away from the 1888-1921 period once, at the end, knowing he will not return.
Since Jonas never traveled between being the Stranger (1888) and Adam (1921), the scars appeared at a time where he wasn't traveling. But Adam does claim that his scars are "because time travel takes its toll".
We initially interpret that as meaning "I travelled so much and all I got were these lousy scars", but realizing that he never travelled during the period in his life where he acquired the scars actually suggests that the scars were caused by him inventing his machine, rather than actually travelling.
That new interpretation still fits with Adam's "time travel takes its toll" claim, albeit a more cryptic way of saying "inventing time travel takes its toll". Adam is known to be intentionally cryptic so it's not out of character for him to be indirect about the source of the scars.
Another small consideration: Adam's scars aren't too different from the abducted children's burnt eyes, and afaik Noah and Adam build a similar electrical device. It's likely that the failed attempts left electrical burns in both development processes.