I should have clarified that i am in the U.S. Here, we would never say "I wrote an application" to mean "I applied for some position". We might say, "I wrote up an application", "I submitted an application", "I filled out an application", et al.
The two candidate meanings for "I wrote an application" here are:
"I authored a software application."
"I authored an application form that others will fill in."
The latter is quite a stretch but it is at least plausible in some scenarios.
"I wrote an application" cannot mean "I filled out an application form" here.
I know I'm late here but I wouldn't use this for filling out an application form in Swedish, either - that would be fyllde i, so a direct cognate.
This phrase, in the context of job applications, I would presume would be about somebody actually writing an application letter, or at least some kind of body of text.