They asked you to give the talk, so therefore it's an invited talk.
In general, what counts as an invited talk can be rather fuzzy, which is why questions like this are valuable. There are clear cases: if you receive an email from the organiser of a seminar or colloquium series at an external organisation, asking you to come and give a talk, then that's an invited talk. Similarly if they ask you to make a research visit to their institution, and give a talk while you're there. But what if you organise the visit, as in this question, and they ask you to give a talk during your visit? This is less clear, since it may well be that they would not have issued the invitation otherwise (although this is always subject to externalities: perhaps they wouldn't have had the funding to invite you, but are delighted that you will be there, since they would have invited you if they could). My view is that in such situations, a clear invitation to give a talk having been issued by the organiser, such talks count as invited.
Even further along this continuum, many contributors to this site appear to agree that even if you had suggested that you could give a talk, it would still count as an invited talk.
JeffE puts this view very clearly in his answer to the linked question:
If you discover that a friend in a distant city is having a birthday party, and you ask "Hey, can I come?" and they say "Sure!", you've been invited to the party.
Same thing goes for talks. When the host institution agreed to let you talk, that was your invitation, which makes it an invited talk.