DO NOT Purchase Invitation Travel Air Tickets for Yourself!
The Reimbursement of What You Pay will be the Subject of the 20.42% Withholding Tax.

Any payment to the invited researchers from host institutes in Japan is the subject of the Japan Income Tax Act.

Be mindful of what you pay in the invitation travel expenses;
If you reserve and purchase the international air ticket for your travel to Japan...
If you reserve hotel(s) and settle payment for your stay in Japan...
If you pay the transportation fees from arriving airport to the venue...

You would most certainly bring payment receipts of the above to the host universities or research institutes for reimbursement, but 20.42% of the total amount will be deducted as the withholding tax irrelevant to its payment method; in cash or wire transfer.

Be advised all payment from universities or research institutes in Japan to you (short-term residents/temporary visitor/non-residents) is the subject of income-tax withholding.

Payment from the Host to Travel Agencies or Hotels is Tax-Free

When your host institute reserves and pays the expenses of your invitation travel directly with the travel agencies and hotels, the payment is not the subject of the income tax. Your host provides you with the flight e-tickets, hotel vouchers and likewise, train/bus tickets for your ride from the arriving airport.

Honorarium and Daily Allowances are Handled According to the Tax Convention

If your host offers you honorarium and/or daily allowances, the payment would be made either in the following two ways;

1) in JPY cash on your lecture day.
2) by wire transfer to the bank account of your choice after the meeting.

• ・Wire transfer to the U.S. Government officials must be made to the designated bank account of the U.S. Department of the Treasury (Foreign Corrupt Practices Act 1977; FCPA).
• ・The similar rules are set for the U.K. Government officials in the UK Bribery Act 2010 (UKBA).

In whichever way the payment should be made, you will have to prepare documents to apply income tax convention and mail them BEFORE you arrive in Japan.
Citizens from certain nations need to hand in the resident certificate issued by the local government. For example, the U.S. citizen has to attach the resident certificate issued by the IRS, and it could take 60 days to issue.

You will have to arrange relevant documents and mail them to the host BEFORE you arrive in Japan.APPLICATION FORM FOR INCOME TAX CONVENTION
https://www.nta.go.jp/taxes/tetsuzuki/shinsei/annai/joyaku/annai/pdf2/256.pdf