To have a quick display name you could also pick a name from the pool, which has the sence in your game world. For example, pick one randomly from the pool [Dragon, Knight, Golden Sword, ...].

As the name is given anyway (either by guessing or by picking), then options for choosing display name could be moved to edit profile screen, so you make connection more faster. The proposed state diagram is:

Now optional Edit profile is available on Connection screen, as shown in your sketch.

EDIT
It is important note of 95% users, which are satisfied with option #4. So you could try to combine #2 and #4 in a way, when name is displayed in input field:

• the guessed name is visible enough for user to take his attention
• a user can change it easily if he dislikes it

To have a quick display name you could also pick a name from the pool, which has the sence in your game world. For example, pick one randomly from the pool [Dragon, Knight, Golden Sword, ...].

As the name is given anyway (either by guessing or by picking), then options for choosing display name could be moved to edit profile screen, so you make connection more faster. The proposed state diagram is:

Now optional Edit profile is available on Connection screen, as shown in your sketch.

EDIT
It is important note of 95% users, which are satisfied with option #4. So you could try to combine #2 and #4 in a way, when name is displayed in input field:

• the guessed name is visible enough for user to take his attention

• it is clear that the name is editable

• a user can change it easily if he dislikes it

To have a quick display name you could also pick a name from the pool, which has the sence in your game world. For example, pick one randomly from the pool [Dragon, Knight, Golden Sword, ...].

As the name is given anyway (either by guessing or by picking), then options for choosing display name could be moved to edit profile screen, so you make connection more faster. The proposed state diagram is:
![enter image description here][1]

Now optional Edit profile is available on Connection screen, as shown in your sketch. [1]: https://i.sstatic.net/rgxKV.png

To have a quick display name you could also pick a name from the pool, which has the sence in your game world. For example, pick one randomly from the pool [Dragon, Knight, Golden Sword, ...].

As the name is given anyway (either by guessing or by picking), then options for choosing display name could be moved to edit profile screen, so you make connection more faster. The proposed state diagram is:

Now optional Edit profile is available on Connection screen, as shown in your sketch.

EDIT
It is important note of 95% users, which are satisfied with option #4. So you could try to combine #2 and #4 in a way, when name is displayed in input field:

• the guessed name is visible enough for user to take his attention
• a user can change it easily if he dislikes it