Both credit to and credit with have the meaning of attribution. The difference is that credit to emphasizes attribution and credit with emphasizes belonging.

credit to

Attribute; attribute... to (someone or something):

The shortage of rice was credited to lack of rain.

The rice harvest failed due to lack of rain.

Credited:

She credited $100 to my account.

She credited $100 to my account.

2.credit with

credit [also credit to]:

She credited herself with the discovery.

She credited herself with the discovery.

She takes credit for that discovery.

To credit [also credit to]:

She credited your account with the amount of the cheque.

She credited the amount of the check to A credit to your account.

Think or believe that (someone) has (a certain quality or achievement, etc.):

He came to the firm already credited with many honours.

He came to the firm already credited with many honours.

He came to the firm already credited with many honours. When I came to the company, I already had a lot of reputation.

3.credit

n. Credit, reputation; [financial] loan; credit; trust; reputation

vt. Believe, trust; attribute to... , attribute to; praise

credit card?credit card; charge card

on credit?credit