The behavior you are experiencing seems to match the documentation. It explains that if you disable prorations (by using proration: 'none'), a new invoice will be generated for the full amount, which could lead to overcharging your customer.

After you reset the billing cycle anchor, Stripe immediately sends an invoice. Enable proration to credit the customer for any days already paid in the previous period. Disabling proration might result in overcharging your customer.

This behavior appears to be specific to subscriptions created in classic mode.

However, with subscriptions created in flexible mode, the desired result will be achieved (using the same Update Subscription code), with no invoice generated and the billing cycle reset to 'now'.  I managed to test this with test clocks on my account and got the same results.

I would suggest that you create Subscriptions in flexible mode or migrate your Subscriptions to flexible mode. However, if your Subscriptions are required to be created in classic mode, you can try changing the billing period by using a trial period, though this will require a few additional steps listed here.