TINYINT uses as you mention yourself a single byte of storage. In SIGNED mode this represents -128 to 127 and UNSIGNED 0 to 255. However, using TINYINT(1) does not alter the storage capacity used in the database.
The display width (1 in TINYINT(1)) does not constrain the range of values that are stored in the column.
Here is an extract from the MySQL manual:
The display width does not constrain the range of values that can be stored in the column. Nor does it prevent values wider than the column display width from being displayed correctly. For example, a column specified as SMALLINT(3) has the usual SMALLINT range of -32768 to 32767, and values outside the range permitted by three digits are displayed in full using more than three digits.