申し込み and 受付: applying and receiving applications
Distinguish an applicant's request from the process or desk that receives it.
The applicant's action
申し込み is read もうしこみ and refers to applying for a course, service, or similar offering. 「講座の申し込みをしました」 means that the speaker applied for the course, not that they have already attended it.
As a verb, use 申し込む, as in 「講座に申し込みます」. The course being applied for takes に, while a form being submitted can appear with を in 「申込書を出す」.
The receiving process and location
受付 is read うけつけ and refers to receiving applications or visitors, or to the desk that does so. 「受付は九時に始まります」 announces that registration opens at nine.
In 「受付で名前を伝えてください」, 受付 is the reception desk where you should give your name. Replacing it with 申し込み would lose that location meaning.
Reading completion messages precisely
「申し込みの受付が完了しました」 means that the application has been received. If screening or a lottery follows, this sentence alone does not confirm acceptance or selection.
「申し込みはオンラインで、当日の受付は会場で行います」 separates online applications from check-in at the venue on the day. The two words can refer to different stages of the same event.
What do the two words mean in 「受付で申し込みができます」?
Show the answer and why
受付 is the desk where you can apply, and 申し込み is the application itself. The sentence says that you can apply at reception; it does not say that you have already applied.