- To read a private blog.
- To become an author.
- To become an administrator, or maybe the new owner of the blog.
To accept membership, you are required to provide a Blogger account, or to setup a new Blogger account. You should see a version of the standard Blogger / Google login display. There are several possibilities, when you use the Blogger / Google login wizard.
Note that the invitation - to a private blog or a team blog - has an expiration date. If you want to access the blog, you are supposed to accept membership when you open the email (or view the blog once) - you should not use the invitation itself as a token for ongoing access. The stated guest membership period may, or may not, be accurate.
The login process will not verify for you, that you are using the correct Blogger account - that is your responsibility. Always use the correct account, when accepting membership.
When you wish to accept blog membership using a Google email address, you provide the password for that account, and the password is verified against the account. This verifies that you own the account in question. Blogger / Google accounts based upon GMail accounts are authenticated, and account ownership is verified.
When you wish to accept membership in a blog using a Blogger account based upon a non Google email address, you provide the password for the Blogger account, and the password is verified against the Blogger account (for an existing Blogger account), or a new Blogger account is setup. In neither case is ownership of the non Google email address verified, nor is the password checked against the non Google email account.
You are allowed to accept membership using any Blogger account that you may have, or may create - regardless of the email address used in sending the invitation, by the blog owner. If you accept membership using an email address / Blogger account that does not appear in the original Permissions based invitation, your membership acceptance should, even so, be accepted - provided the mysterious "100 members, total" limit is not exceeded.
Possibilities for more problems come later.
When you change the password for your Google / GMail account, the password for the Blogger account changes too - since it's really one account. When you wish to change the name of the GMail / Google account, you have to setup a new account, and transfer all assets to the new account. The latter process is clumsy, and complex - but whether you change the account name, or the password, you know what you have done. And the account name, and password, is always synchronised.
When you change the password for a non Google account, that's all that changes. If you have a Blogger account that's based upon a non Google email address, that password does not change - unless you change it intentionally, and separately. When you rename a non Google email account, or setup a new non Google email account to replace an existing account, again, your Blogger account does not change.
If you later become confused about a Blogger account name or password, where the Blogger account is based upon a non Google email address, you will require access to the non Google email account, to recover the Blogger account or reset the Blogger password. Lack of access to the non Google account, whether it was yours in the first place - or whether it never existed in the first place, will leave you unable to manage your blog.
As always, you are responsible for using the right Blogger account when accepting membership - just as you are responsible for proving your identity or membership, in general.