We were taught a third method: have the side of the cello towards the floor, and lay the instrument against the edge of your chair, in a diagonal position. The top of the waist of the cello gets 'seated' on the chair pad, and the end pin is out, in its hole (just when it is played). The rib edges and the pin together will stabilise the cello, if it is laid properly. The bow gets to be laid on the seat.
This way the cello is occupying less than half of floor-space (as opposed to be layed on the floor), and it is both stable and very orderly. Also this is very swift, and takes less movement / manouevering by the player.
This is a very orchestral approach, as it is most important in that setting that the players are swift, and are not only orderly, but also orderly looking. They are learning and practicing to move in unison in a very orderly way not only when playing, but also when they need to leave their instruments behind (eg when the show starts, or is over, or goes for an interval). The players and orchestra practice this all the time (to move together, be orderly, be neat looking, etc), so these rules go on in practice times as well, and layers are taught to do the same at home.
It is part of the cello practice and cello culture to move around the instruments with care and take caution, so it becomes second nature.
Cellos do not belong in their casings all the time, it is just simply not practical. Dragging the cello back / forth and in / out of its casing unnecessarily is poor practice, and actually increases the chances of accidents. Putting in the end stick and pulling it out each time the group goes on a break is also poor practice. It requires double / triple handling, it is simply not economical, not a good use of energy, and otherwise unpractical.
If your orchestra group laid down each cello with its side against the chair, you would have more room. If the chairs and cellos were lined up as well (in an orderly way) and adequate space between them, then the group would eventually find it easy to not bogged in with feet, loose clothing, etc, not to trip / knock the instruments over - it will become a house rule, as it should.
It is the responsibility of the conductor, or team leader / coach, to set these rules, and to keep them going. These rules are also necessary for occupational safety of the players (not only for the cellos), as someone tripping over a cello could end up breaking a leg, or hand, or neck (compared to that, the falling of the bridge would not be a big deal!).
If you do not have a regular coach, you need to agree amongst yourselves about how to set, and how to keep these rules. Good Luck.