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I do not think the one click update works but I haven't tested it, just guessing from what other forum members are posting.
If you don't need modules, and you shouldn't since most modules are not prestashop 9 ready, I think these are the 73 sql tables you want for a complete migration:
ps_accessory
ps_address
ps_carrier
ps_carrier_group
ps_carrier_lang
ps_carrier_shop
ps_carrier_tax_rules_group_shop
ps_carrier_zone
ps_cart
ps_cart_cart_rule
ps_cart_product
ps_cart_rule
ps_cart_rule_lang
ps_category
ps_category_group
ps_category_lang
ps_category_product
ps_category_shop
ps_cms
ps_cms_category
ps_cms_category_lang
ps_cms_category_shop
ps_cms_lang
ps_cms_shop
ps_contact
ps_contact_lang
ps_contact_shop
ps_customer
ps_customer_group
ps_customer_message
ps_customer_thread
ps_delivery
ps_employee
ps_employee_shop
ps_image
ps_image_lang
ps_image_shop
ps_import_match
ps_lang
ps_mail
ps_manufacturer
ps_manufacturer_lang
ps_manufacturer_shop
ps_message
ps_meta
ps_meta_lang
ps_order_carrier
ps_order_cart_rule
ps_order_detail
ps_order_detail_tax
ps_order_history
ps_order_invoice
ps_order_invoice_payment
ps_order_invoice_tax
ps_order_payment
ps_order_state
ps_order_state_lang
ps_orders
ps_pack
ps_page
ps_page_type
ps_product
ps_product_lang
ps_product_sale
ps_product_shop
ps_product_supplier
ps_product_tag
ps_stock_available
ps_supplier
ps_supplier_lang
ps_supplier_shop
ps_supply_order_state
ps_supply_order_state_lang
I will post more info once I'm done (I'm migrating to prestashop 9 from 1.7.8.8)
I did not migrate everything at the end of the day. I only got the products.
_accessory,
_category, _category_group, _category_lang, _category_product, _category_shop,
_image, _image_lang, _image_shop,
_product, _product_lang, _product_shop,
_pack
were the only tables I needed for this. I used the prestools tool to remove nulls, dragged and dropped my img folder, regenerated thumbnails and search index and it all worked.