Welcome to the community, @Lauren_Lundholm!
Sadly, Airtable doesn’t offer this capability.
You could do it with JotForm, which offers native Airtable integration (and offers so much more flexibility than what Airtable offers).
Alternatively, if you wanted to do it natively in Airtable, you would need to create additional fields in your table that are “temporary fields” that only exist on certain forms. You would need to make those fields REQUIRED on your forms, so that when somebody submits a form, the VERY PRESENCE of information in that temporary field lets you know which form was submitted.
This temporary field could be as simple as a checkbox that says: “Check this box before submitting the form.” Or, it could even be a field where you collect real information in it — and then you could create some automations in Airtable to move the real information from the “temporary field” into the “permanent field” where that information needs to live.
It’s a big hassle all the way around, and I would highly recommend sending an email to support@airtable.com to request that they please improve their forms in the way that you asked about. So many people are in need of this.