Which channel gets credit for a visitor: the first one or the one of the visit?
It depends on the question the screen answers. Screens about a person — the Visitors tab, the visitor detail and the Navigation tab of the contact record — use the first click: the channel of the first visit, which is what brought that person to you. Screens about traffic — the Analytics tab, with Sessions by channel, Conversion by channel and Campaigns by channel — count each visit under the channel it came through.
Each screen states the model it uses, so you do not have to guess.
An example
Someone sees an Instagram ad, lands on your site and leaves. A week later they search your name on Google, come back and submit a form.
Both numbers are right: one answers "what brought this person", the other "where did the visits come from".
Why the first click on person screens
When you open a visitor or a contact, the question is usually "where did this person come from". The first visit answers that. The later visits stay visible — each with its own channel — in the visitor detail and in the contact record.
Why the visit channel in the report
In the report, the question is "which channel is bringing traffic now". If someone who arrived through Instagram months ago comes back through Google today, today's visit belongs to Google — and the report has to show that so you can see the effect of a new campaign.
"Became a contact" and conversion rate
In both models, a visitor became a contact when they identified themselves (submitted a form, started a checkout or something equivalent). Conversion is always contacts ÷ visitors of the slice: of the channel in Conversion by channel, of the channel + campaign pair in Campaigns by channel.
What does not exist yet
You cannot switch the report to a "last click" model (the channel of the visit in which the person identified themselves) yet. Today the report counts each visit, and the person screens use the first click.