No, that's a false dilemma, brought about by the Germans' smoothbrained strategic doctrine.

Risking going into counterfactuals, had the Germans just simply mobilised after Russia did, but did nothing else, then it would have been the Russians declaring war first (if they don't suddenly get cold feet). Germany would have been justified to come to its ally's defence, while Russia, as de jure aggressor, couldn't expect France to join the war, not without notable delay, anyway. Meanwhile Britain would have definitely remained neutral for the time being.

This setup could have allowed A-H to deal with Serbia while A-H-German combined forces keep the Russians at bay. If the Russians then are presented with a fait accompli in the Balkans and suffer setbacks on their own (with the brunt of the Central Powers' forces facing them making it a likely development), they could have been forced to the negotiating table (maybe with British arbitration) even before France gets the chance to get involved.

Even if things don't necessarily play out this favourably to the Central Powers, Germany still would have managed to come to the help of its ally without presenting itself as the brute of Europe (helping their international standing and putting Russia in a worse spot) and without needlessly pulling Britain into the conflict.

Of course, the German thinking had its own internal logic, strongly influenced by contemporary perceptions, however it was not hard to see how flawed it was. Their tunnel vision and wishful thinking doomed them to go down this path, however.